# Questions tagged [l-functions]

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

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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: L(f;s_1,\cdots,s_n)=\int_0^{i\...
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### Most General Definition of an L-Function

I have seen examples given of $L$-functions, such as Dirichlet $L$-functions and the Riemann Zeta Function, but I have not seen a definition of the most general form of an $L$-function. Basically what ...
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