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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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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
43 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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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
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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64 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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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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54 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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63 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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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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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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74 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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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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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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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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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 ...
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1answer
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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- ...
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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
196 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
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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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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
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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
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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
130 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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1answer
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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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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
64 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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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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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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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 ...
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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
121 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" ...
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1answer
59 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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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}\...
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Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form

Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
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1answer
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Sequence related to degrees of elements of the Selberg class

Say an increasing arithmetic sequence $ (s_n)_{n\ge 0} $ is 'sensible' if every element thereof is the degree of some function belonging to the Selberg class. Let $ a_s $ its reason and $ f_s : =1/...
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1answer
131 views

what does $L(s, \pi \times \chi)$ mean in analytic number theory?

Given a Dirichlet character -- a character $\chi: (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{C}^\times$ one can define a Dirichlet L-function: $$ L(s, \chi) = \sum \frac{\chi(n)}{n^s}$$ if $\pi$ is ...
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1answer
45 views

Eigenforms and L-functions

I'm trying to show the following identity: If $f=\sum_{n\geq 1}a(n)q^n \in S_k$ is a normalized Heckeeigenform, where $k$ can be written as a sum of two even numbers $k= k_1+k_2$, and $L(f,s)=\sum_{n\...
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35 views

Extended Selberg class and RH

Is it known whether an element of the extended Selberg class that fulfills the analogue of the Riemann hypothesis actually belongs to the Selberg class ? Thanks in advance.
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What do $L$-functions of curves over $\mathbb Q$ tell us about the curve

Following up this thread: $L$-function of an elliptic curve and isomorphism class I'd like to ask some more questions for the case of smooth projective curves $C$ over $\mathbb Q$ To be more precise,...