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Questions tagged [l-functions]

L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.

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Relationship between $\theta$ functions and number fields.

I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to. Firstly, I know that $\theta$-functions arise as sections of ...
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36 views

Gamma Function Integral Identity

On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function: $$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
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Expression of the dirichlet coefficient of an L-series in terms of the Satake parameters

Last year, I had found a pdf where the expression of the Dirichlet coefficient $\lambda_{\pi}(p^{\nu})$ in terms of the Satake parameters $\alpha_{p,i}(\pi)$ was given. Unfortunately I don't remember ...
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54 views

Asymptotics for average of Fourier coefficients of cusp form

Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function $$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$ of a weight $k$ ...
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The argument of L-functions

For Riemann Zeta function, we can define $\pi S(t)=\arg\zeta(\frac{1}{2}+it)$ where the argument is the variation from $+\infty+it$ to $\frac{1}{2}+it$. I read few results related to $S(t)$ such as $S(...
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Proof of Chebotarev Density Theorem without Artin Reciprocity

I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity. As I understand it, the main ...
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35 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
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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 $$ ...
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2answers
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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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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 ...
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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 ...
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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 ...
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25 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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78 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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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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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
21 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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71 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
36 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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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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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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276 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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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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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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29 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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79 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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43 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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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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1answer
89 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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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- ...
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1answer
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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(\...
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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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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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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
240 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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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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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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56 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 > ...
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1answer
91 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)...
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1answer
141 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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179 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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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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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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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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29 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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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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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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42 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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239 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?