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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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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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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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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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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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160 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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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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39 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,...
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148 views

Definition of the Weil Group: where is this map $G_1/G_1^c \rightarrow \overline{G}/\overline{G}^c$ coming from?

The following comes from a 1971 paper by R. Langlands, "On the Functional Equation of the Artin L-functions" There seem to be some formatting errors, for example $\mathcal S$ should be replaced by $\...
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223 views

Functional equation of Rankin-Selberg zeta function

Can someone explain to me where come from the functional equation (1.6) in this article ?
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229 views

Dirichlet class number formula

The way the class number formula for Dirichlet L-functions is always proved in modern textbooks and notes is You prove the general class number formula for $\zeta_K(s)$. You prove that for quadratic ...
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Can the expression appearing in Selberg's orthonormality conjecture be viewed as an inner product of class functions?

Selberg's orthonormality conjecture states that if $F$ and $G$ are primitive functions of the Selberg class, then one has: $\displaystyle{\sum_{p\leq x}\dfrac{a_{p}(F)\overline{a_{p}(G)}}{p}}=\delta_{...
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Unicity of Rankin-Selberg 'factors'

Suppose $H$ is an automorphic L-function obtained by applying the Rankin-Selberg convolution to two automorphic L-functions $F$ and $G$ each of them being different from the Riemann Zeta function. Is ...
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Kernel affine space of an L-function

Throughout this question, an L-function is both an automorphic L-function and an element of the Selberg class such that whenever $F$ and $G$ are L-functions, then so are $F.G$ and $F\otimes G$, where $...
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Determine Fourier coefficients by the values of its L-Series.

suppose I know the values of $\sum_{n=1}^\infty \frac{a_n}{n^k}$ for all $k=1,2,...$. Is there a way/tool to determine the coefficients $a_n$ from this (which might not be unique)? I would appreciate ...
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Showing $L(1,\chi)$ is positive given that it's nonzero

Let me first provide context for this question. There is a series of four exercises in Ireland & Rosen's book (in second edition it's exercises 14-17 in chaprer 16), aim of which is (although ...
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L-series through integrals of rational functions

Recently I stumbled upon this short proof here: $$L(1,\chi_2)=\sum_{j=0}^{+\infty}\left(\frac{1}{3j+1}-\frac{1}{3j+2}\right)=\int_{0}^{1}\frac{1-x}{1-x^3}\,dx=\int_{0}^{1}\frac{dx}{1+x+x^2}$$ so: $$\...
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How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
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60 views

Integral formula for the local L factor of a base changed automorphic representation

Let $\Bbb A$ the ring of rational adeles and let $\pi=\bigotimes_{p\leq\infty}\pi_p$ be an automorphic (cuspidal) representation of ${\rm GL}_2(\Bbb A)$. Fix a quadratic extension $K\supset\Bbb Q$. ...
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What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where L-...
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76 views

L-functions identically zero

Let $f:\mathbb{N}\to\mathbb{C}$ and assume that the L-function $$L(f;s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$$ converges absolutely on some right half-plane $Re(s)>k$. Is it true that $L(f;\cdot)$ ...
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Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such ...
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Covariance group of the functional equation of an L-function

These last few days, I've been wondering whether one could consider the parameters/variables $\chi$ and $s$ a Dirichlet L-function depends on as coordinates such that the pair of transformations $(\...
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367 views

What do the zero's of L-functions entail?

I don't know exactly how, but I've read the Riemann Zeta function's nontrivial zero's imply something about an error term for an approximation function thing for the Prime Counting Function. I found ...
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Does $F\otimes G\in\mathcal{M}$?

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ ...
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Analytic formulas for special values of $L$-functions

In The Princeton Companion to Mathematics, IV.1, “Algebraic Numbers”, the conditionally convergent series \begin{equation} (1)\qquad\frac{\log(\sqrt{2}+1)}{\sqrt{2}}=1-\frac{1}{3}-\frac{1}{5}+\frac{1}...
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Sum of reciprocals of primes diverges

I can show that $$\log(\zeta (s)) = \sum _{p\in\Bbb P} \frac{1}{p} + R(s)$$where $$R(s) = \sum _{m\geq 2} \sum_{p\in\Bbb P} \frac{1}{m} \frac{1}{p^{ms}}$$ where $\Bbb P$ is the set of all primes, ...
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elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
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is this map necessarily a field automorphism?

Let $M$ denote the intersection of the Selberg class and the class of automorphic L-functions and let's define the automorphism group of $M$, denoted by $Aut(M)$, as the group (under composition) of ...
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dimension of a scheme and degree of an L-function

Disclaimer: I first asked this question on Mathoverflow but I was told it was off-topic for that site, so I ask it here. I try to understand correctly the notion of scheme, as Serre in the second ...
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definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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Effective bounds on L(1,Chi) for Chi a Dirichlet Character

I have $\chi$, a Dirichlet Character $\bmod n$, and I have established that $L(1,\chi) \geq C / \log \log n$ for some constant C under the generalized Riemann Assumption. I'll call this proposition $A$...
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Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help ...
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a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
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On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
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expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
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Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{...
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Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
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Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly appreciated....
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443 views

$L$-function of an elliptic curve and isomorphism class

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same $...
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610 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
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Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
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Why are L-functions called such? [duplicate]

L-functions are discussed here: http://www.lmfdb.org/intro/tutorial... So why are they called L-functions...? Is there a reason for the L?
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Dirichlet L function

The function is defined here - http://en.wikipedia.org/wiki/Dirichlet_L-function If $\chi$ is primitive and $\chi(-1)=1$ how do I show that $L$ has infinite number of zeros in the critical strip
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Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In SAGE...
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On the special value of Hecke L function.

For a nontrivial Hecke character $\chi:A_Q^{\times}/Q^{\times}\to S^1$, we know $L_Q(s,\chi)$ is nonzero. Is this true for number field $F$? I know is is holomorphic at $s=1$ by Artin conjecture, but ...