Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
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Need reference for proof of functional equation for generalized L-functions
When proving functional equation for Riemann zeta function one starts at the definition of gamma function
$$\Gamma(s) = \int_0^{\infty} x^{s-1} e^x\mathrm dx\tag1$$
After a few steps we arrive at
$$ ...
2
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2answers
63 views
Test vector for local zeta integral with ramified character
Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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0answers
31 views
Langlands L functions for groups over finite fields.
In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
2
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0answers
23 views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
(1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
4
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0answers
79 views
Determining elliptic curve analytic rank even/odd
For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if ...
-1
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1answer
21 views
Permutation group of Satake parameters
Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $ L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $ G_{p}$ of the ...
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2answers
57 views
Evaluating Dirichlet $L$-functions at $s=1$
I'm trying to find references on general methods for evaluating Dirichlet $L$-functions at $s=1$, but it's proving a little harder to google than I'd hoped. Specifically I'm looking for any books or ...
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1answer
44 views
Series of reciprocal of integers
This is a question I asked myself today...
$ $
Do you know if it is possible to build a strictly-increasing sequence $(u_n)_{n\in\mathbb{N}^\star}$ of positive integers such that $\displaystyle\sum_{...
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0answers
29 views
Is $ \pi\mapsto(s\mapsto L(s,\pi)) $ bijective?
Let $ \pi $ be an automorphic representation of $ \operatorname{GL_{n}}(\mathbb{A}_{\mathbb{Q}}) $ and $ L(s,\pi) $ the associated L-function. Is the map $ \pi\mapsto L(s,\pi) $ bijective ? In ...
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0answers
12 views
self-intertwining operators of an automorphic representation
I stumbled a few days ago on the definition of an intertwining operator in https://www.encyclopediaofmath.org/index.php/Intertwining_operator
Considering the case where $ \pi_{1}=\pi_{2}=\pi$, $ ...
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1answer
20 views
Modular parametrization from equality of $L$-functions
In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways
1) if there exists a nonconstant morphism $X_0(N) \to E$,
2) if there exists a modular form $f$ ...
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1answer
67 views
Is the conductor of an L-function F the absolute value of the discriminant ofsome number field related to F?
In the theory of automorphic forms, ramified primes of an L-function divide the so-called conductor thereof. On the other hand, one can define for a number field $ K $ an integral invariant $ \...
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1answer
27 views
Uniqueness of $L$-series of cusp forms
For a cusp form $f$, one gets an $L$-series by taking the Mellin transform as we have
$$ \tilde{f}(s) = (2\pi)^{-s} \Gamma(s) L(s,f). $$
My question is: is this operation injective? It seems to me ...
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1answer
60 views
Convexity Bound of Rankin-Selberg L-Function
Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the ...
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1answer
68 views
Rankin-Selberg convolution : normalization issues
I was told several times on MO that if $ F : s\mapsto\sum_{n>0}\frac{a_{n}}{n^{s}}$ and $ G : s\mapsto\sum_{n>0}\frac{b_{n}}{n^{s}} $ for $ \Re(s)>1 $ are L-functions, then provided the ...
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0answers
52 views
The $\heartsuit$ operator on $\mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$
In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(...
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1answer
189 views
Roadmap to Iwasawa Theory
I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good ...
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0answers
75 views
Upper bound on zero multiplicity of Dirichlet $L$-functions
I was wondering whether there is (a known) upper bound of the order of the non-trivial zeros of Dirichlet $L$-functions. For a zero $s$ of the Riemann zeta function we have the estimate $C\cdot\log(\...
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0answers
26 views
Bounds for truncated $L$-series over short intervals
Let $\chi$ be a non-principal Dirichlet character. Are there any good non-trivial bounds for short sums of the form
$$
\sum_{x < n \leq x + N} \chi(n)n^{i t}
$$
as both $x \geq 1$ and $t \in \...
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0answers
65 views
Simplest nontrivial example of an L-function yielding information about a Diophantine equation
I got excited while reading Langlands' essay, REPRESENTATION THEORY: ITS RISE AND ITS ROLE IN NUMBER THEORY, because he appears to provide concrete motivation for the study of L-functions:
We have ...
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0answers
42 views
Artin $L$-functions and abelianization
Let $L/K$ be a finite Galois extension of global fields, with $G=\mathrm{Gal}(L/K)$. Let $H \leq G$ and $\chi:H \to \mathbb{C}$ a non-trivial irreducible character of $H$. Then we can define the (...
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1answer
83 views
Why does the Dedekind zeta function of a number field have a pole at $s=1$?
The analytic class number formula tells us that the Dedekind zeta function $\zeta_K$ of a number field $K$ has a pole at $s=1$ with residue $$\frac{2^{r_1}(2\pi)^{r_2}\text{Reg}_Kh_K}{w_K\sqrt{|\...
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31 views
Convergence of the Artin L-function
Let $L/K$ be a Galois extension of number fields with Galois group $G$. Let $\rho: G \rightarrow \operatorname{GL}_n(\mathbb C)$ be a representation with character $\chi_{\rho}$. For each unramified ...
4
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1answer
81 views
Why does the determinant come in for Artin L-Functions?
Let $L/K$ be a Galois extension of number fields. For $\mathfrak p$ a prime of $K$, unramified in $L$, the Frobenius elements $\sigma_{\mathfrak P}$ for $\mathfrak P \mid \mathfrak p$ are conjugate, ...
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1answer
55 views
Trying to understand generic nature of L-functions
I’m not a “professional” mathematician but amateur and curious in maths - more like a hobbyist. So this is a little bit advanced topic for me and am trying to visualize/grasp understanding of how L- ...
2
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1answer
74 views
Analytic continuation of twisted Hecke $L$-function
Let $K$ be a (real quadratic) number field and $\chi$ be a Hecke character on it. We can think about Hecke $L$-function
$$
L(\chi, s):= \sum_{0\neq \mathfrak{a}\subseteq \mathcal{O}_{K}} \frac{\chi(\...
3
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0answers
95 views
Understanding a result of Serre about zeros of $x^3 - x - 1$ in $\mathbb{F}_p$
I'm trying to understand a result of Serre which relates the number of zeros in the finite field $\mathbb{F}_p$ of $f(x) = x^3 - x - 1$ to a modular form. The result can be found in the section 5.2 of ...
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0answers
25 views
Reference for $L$-functions of curves
I am looking for a reference that explains as easily and as completely as possible how the $L-$function of a curve $C$ (non-singular, projective, geometrically irreducible, defined over $\mathbb{Q}$, ...
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44 views
Washington's Formula for $p$-adic $L$ function
I am reading through Washington's construction of $p$-adic $L$ functions in chapter $5$ of his book Cyclotomic Fields. Instead of $p$-adically interpolating as Kubota and Leopoldt originally did, ...
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1answer
220 views
Is the Generalized Riemann Hypothesis thought to be true?
The Riemann Zeta Function is the analytic continuation of the following function:
$$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
The Riemann Hypothesis states that the zeros of this in the critical ...
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1answer
105 views
How to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$?
I am trying to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$ where $\chi = \mathbb{A}^\times / \mathbb{Q}^\times$ and $\pi$ is a cuspidal representation of $GL_2( \mathbb{A})$ (where $\...
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0answers
51 views
Study L-function associated to elliptic curves
I'm studying Iwasawa theory, and now I'm reading Greenberg's article about the application of Iwasawa theory of elliptic curves, but I have some problems on the first chapter, because it's about Hasse ...
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0answers
49 views
uniform convergence of $L(s,\chi)$ for $\Re(s) ≥ 1 + \delta$ " due to absolute convergence for $\Re(s)>1$?
On page 6 of this link, lemma 2.4 shows $L(s,\chi)$ is absolutely convergent for $\Re(s)>1$.
I understand the proof. However,
they also add: "The above proof also shows that for any $\delta > ...
2
votes
1answer
86 views
Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $
I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product:
$$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2
= q \prod_{n=1}^\infty (1 - q^n)...
4
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1answer
134 views
Generalization of a $\det$ series for $\zeta_{\mathbb{Q}(\zeta_k)}$
With $\mathbb{Q}(\zeta_k)$ a cyclotomic field, $\chi_1,\ldots,\chi_{\phi(k)}$ the Dirichlet characters modulo $k$ and $\tilde{\chi}_1,\ldots,\tilde{\chi}_{\phi(k)}$ the underlying primitive Dirichlet ...
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1answer
151 views
Bounds for Dirichlet L-functions
In the half plane $\sigma$=Re(s) > 1 , one can find bounds for the Riemann zeta function $\zeta$(s) using either its convergent series or product formula.$\,$ From the Dirichlet series we get the ...
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93 views
Question regarding the number of zeros of Dirichlet L-function
I have encountered the following result: Let $T\geq 2$, and let $N^*(\alpha, q, T)$ denote the number of zeros of all the L-functions $L(s, \chi)$ with primitive characters $\chi$ modulo $q$ in the ...
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0answers
83 views
Why do we study moment of Riemann zeta function and moment of Dirichlet L-function?
I study Analytic Number theory and in particular L-functions. I began to study moments of Riemann zeta function and Dirichlet L-function. However I do not see why this is important, what is the ...
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0answers
24 views
Local parameter of $L(s,f)$ at infinity $k_j$
one of the criteria for $L(s,f)$ to be an $L$-function is if it has a Gamma factor
$\gamma(f,s)=\pi^{-ds/2} \prod_{j=1}^d \Gamma\left(\frac{s+k_j}{2}\right)$
where d is the degree of the Euler ...
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1answer
28 views
On coefficients of L-functions
I consider holomorphic modular forms $f$ of weight $k$ and level $1$. Introduce its associated $L$-function:
$$L(s,f)=\sum_{n=1}^\infty \frac{\lambda_f(n)}{n^{s}}$$
It has an Euler product $\prod_p ...
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0answers
45 views
Show that coefficients of $\eta(z)^2 \eta(11z)^2$ are multiplicative
Can someone help me find the Euler product associated to this newform? I have a product of eta functions:
$$ \eta(z)^2 \eta(11z)^2 = q \prod_{n=1}^\infty (1-q^n)^2 (1 - q^{11n})^2 = q - 2q^2 - q^3 +...
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0answers
37 views
How to induce a morphism between compact sheaf cohomology groups from the map of two space?
I'm reading the proof of algebraicity theorem for L-functions associated with cusp forms from Hida's book.
In this proof,the author related L-function's values with a differential form on the modular ...
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1answer
65 views
Gamma factor of $ F\otimes G$
Let $F$ and $G$ be elements of the Selberg class with respective gamma factors $\gamma_{F}(s)=Q_{F}^s\prod_{i=1}^{d}\Gamma(s/2+\mu_{i}(F))$ and $\gamma_{G}(s)=Q_{G}^s\prod_{j=1}^{d'}\Gamma(s/2+\mu_{j}(...
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220 views
Reference for $L$-functions
What will be a good reference to study $L$-functions for a beginner? Is there any book/lecture note in complex analysis that covers it?
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1answer
30 views
A detailed explanation of why $\zeta(s)$ has a gamma factor in the spirit of the corresponding axiom related to the Selberg class
Wikipedia's article for the Selberg class tell us that the Riemann's Zeta function belongs to the class, thus satisfies the Definition.
I would like to know the details using this terminology of the ...
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0answers
19 views
Proving $L(S,\chi)$ is continuously differentiable for $0<s<1$
I am trying to understand the proof of the fact that the $L-\text{function}$ given by $L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ ($\chi$ is a Dirichlet character modulo $q$) is continuously ...
2
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0answers
62 views
Dirichlet L-function has no zeroes in Re(s) > c implies Euler product converges in Re(s) > c?
If a Dirichlet $L$-function has no zeroes in $\Re(s) \gt c$, does its Euler product necessarily converge in $\Re(s) \gt c$?
So I know the proof that (conditional) convergence of the Euler product $\...
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1answer
126 views
how to construct the Z-function corresponding to Davenport-Helbron L-function?
See https://aimath.org/news/gl3/zfunction.html
Actually, this question doesnt make sense, the Davenport-Heilbronn "zeta function" isn't even an L-series so it isn't right to call it a "zeta function" ...
1
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1answer
90 views
Prove there are more than $n$ prime numbers $p \leq x$ for $x \geq 3$.
Let $x$ be greater than or equal to $3$. Prove there are more than $\frac {\ln(\ln(x))}{\ln 2}$ prime numbers $p \leq x$.
Hint: use Euclid's proof and induction.
I have run across this question in a ...
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0answers
17 views
Does this map send a member of the Selberg class to another one ?
Let $ F $ be an element of the Selberg class of degree $ d>1 $ and let $ p $ a prime dividing $ d $ . Does the map $ \phi_{p} : a_{n}(F)\mapsto a_{n}(F)^{1/p} $ where $ F(s)=\sum_{n>0}\...
