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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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### Normalisation of $L$-function for classical modular forms and automorphic representations

I found that the normalisation of $L$-functions of classical modular forms and corresponding automorphic representations is somewhat confusing for me. Recall that if $f\in S_k(\Gamma_0(N))$ is a ...
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### A question on the Riemann zeta function

Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ... 47 views

### Approximate functional equation of Selberg Class L-functions

I was reading the paper "Integral moments of $L$-functions" by Conrey et al. They have used the following sharp cutoff version of the approximate functional equation for "Selberg class&...
1 vote
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### Whittaker at Archimedean Test vector

Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression ...
1 vote
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### Why are Rankin-Selberg convolutions different when $n=m$?

Let $\pi$ an automorphic cuspidal representation on $GL_n(\mathbb{A})$ (and similarly $\pi'$ on $GL_m$). For $m=n-1$, it is standard to introduce the Rankin-Selberg L-function as (the gcd of the ...
1 vote
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### What is the importance of RH for $L$-functions of modular forms?

The Reimann Hypothesis (RH) for $L$-functions of modular forms states that all the non-trivial zeroes of an $L$-function of a modular form must lie on the critical line. My question is: why is this ...
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### *Why* the converse theorem (conjecture) should be true?

Currently, the following $\mathrm{GL}(n)$ converse theorem due to Cogdell and Piatetski-Shapiro is known: For an admissible irreducible representation $\pi$ of $\mathrm{GL}(n, \mathbb{A})$, $\pi$ is ...
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### Logarithmic average of the Legendre symbol

My question is simple: can we show that the sum $$\sum_{k=1}^{p-1} \frac{\left( \frac{k}{p} \right)}{k}$$ is positive for all primes $p$ where $(k/p)$ denotes the Legendre symbol modulo $p$, i....
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### Relations between different zeta functions for a simple algebra

I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
1 vote
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### Triangle inequality of Dirichlet cofficients of Artin L-function.

As for automorphic L-function we know that $$\left|\lambda_{\pi}(n)\right|^{2} \leq \lambda_{\pi \times \widetilde\pi}(n)$$ is there exist a inequality similarly for Artin L-functions? For above ...
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### Why is $s=1$ in Dirichlet's Class Number Fomula when $L$-functions are only defined for $\Re(s)>1$?

Given a quadratic number field $K$, let $w$ be the roots of unity, $d_K$ the discriminant, $h$ the class number and let $\varepsilon$ be the fundamental unit of $O_K$. We have that Dirichlet's Class ...
1 vote
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### Proposition 10.4 in Neukirch's Algebraic Number Theory

I am stuck on a small detail in the proof of Proposition 10.4(iii) from Chapter VII of Neukirch's Algebraic Number Theory. For a Galois extension of number fields $L|K$ and a representation $(\rho, V)$...
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### Residue of Rankin Selberg L-function

Let $f$ be a normalized holomorphic cusp form with weight $k$, level $N$. The Fourier expansion of $f$ can be written as \begin{align*} f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{(k-1)/2} e^{2\pi inz} \...
1 vote
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### Meromorphic continuation of L-functions

I am following these notes and on page 2 the claim is that if we have an $L$-function $$L(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ with $a_n=O(n^r)$ and if $L$ has a meromorphic continuation and ...
1 vote
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### Approximate functional equation of the L function

I was reading a paper on Selberg's central limit theorem for the classical automorphic $L$ functions attached to primitive holomorphic cusp form $f$. I can not understand the following equation. \...
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### Factorization of $L$-functions for CM Elliptic Curves

I saw recently that the $L$-functions of elliptic curves with CM can be factored as a product of simpler $L$-functions. In this question, I'd like to ask why that factorization is significant and what ...
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### Searching for a concept/clue: efficient algorithm to extract columns in a large table/matrix with minimal collisions: a use case of class field theory

Although the algorithmic problem is very generic in nature and there are many (possibly more down to earth) examples of its application, I do not want to suppress the real context of my problem. I ... 85 views

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### Dirichlet L-series and Hecke L-series

I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?
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### L-function of Hilbert modular forms of non-parallel weight

I found that in many literature, they only define the L-function of Hilbert modular forms of parallel weight. Can we define the following L-function: \begin{equation} L(f;s_1,\cdots,s_n)=\int_0^{i\...
1 vote
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### Definition of $L$-function of elliptic curves

$L-$function of elliptic curves is Dirichlet series and defined to be $$L(E,s) = \sum_{n\ge 1}\frac{a_n}{n^s} = \prod_p L_p(E,s),$$where the Euler factor at $p$ is  L_p(E,s) = \begin{cases}(1-a_pp^...
1 vote
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### Limits with zeroes of L-functions

As usual, let $L(s,\chi) = \sum_{n\geq 1} \chi(n)n^{-s}$ be the L-function of a Dirichlet character mod $q$ and assume GRH. Since no nonzero meromorphic function has infinitely many zeroes in a ...
### Special values of Hecke $L$-function on imaginary quadratic fields
Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Let $\chi$ be an algebraic Hecke character on $K$ with conductor $\mathfrak{f}$ and infinity type $(a,b)$, i.e. ...
### Question on paper of Mazur, Tate, Teitelbaum and $p$-adic L functions of modular forms
I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following. Let $f$ be a cuspidal eigenform of weight $k$ and ...