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&...
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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 L-factors polynomials in $q^{-s}$?
I am interested on integrals of "Tate style", say of the form
$$\int_{F^\times} \Phi(a) \kappa(a) |a|^s d^\times a$$
Here, $F$ is a local field, $|\cdot|$ the associated absolute value, $\...
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Davenport, the usage of mean value theorem
In Davenport, Multiplicative Number Theory, page 6, the author asserts that:
"Let $\omega$ be a complex character. Suppose $L_{\omega}(1)=0$, then it would imply that $L_{\omega}(s)=L_{\omega}(s)-...
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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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Looking for table of special values of the Dirichlet $L$-function
For double checking calculations I made I'd like to find a table of values of $L(-1,\chi_D)$ for small positive fundamental discriminats $D$. It there a table somewhere in the internet? Where?
With $\...
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Artin $L$-function is well defined
Here is what I got so far. Let $\rho: Gal(L/K)\rightarrow GL(V)$ be an Artin representation. Let $\frak{p}$ be a prime of $K$, $\frak{P}$ a prime of $L$ lying above $\frak{p}$ and denote by $D_\frak{...
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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}
\...
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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 ...
user736865
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Controlling High Moments with Short Low Moments
Recently I've been working on a project for which I've needed to reference Iwaniec's paper Fourier coefficients of cusp forms and the Riemann zeta function, where the short fourth moment estimate
$$
\...
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Adapting the proof of $L(\chi,s)\ne 0$ for $R(s)=1$ for number fields on function fields
I'm looking at the proof of JP Serre on number fields showing that the $L$-serie $L(\chi,s)$ is non vanishing on the line $Re(s)=1$ if $\chi$ is a non trivial character. I try to adapt it for Hecke ...
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Explicit computation of the factor in the interpolation formula for p-adic Rankin-Selberg L functions
I'm currently trying to understand the appendix of this article by David Loeffler https://arxiv.org/pdf/1704.04049.pdf, which consists in an explicit computation of the factor allowing the ...
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Functional equation for Rankin-Selberg L functions in the imprimitive case
If $f$ and $g$ are primitive modular forms of characters $\chi$ and $\psi$, such that $\chi, \psi$ and $\chi * \psi$ are all primitive, then we have an explicit functional equation. This is proven in ...
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Is there a Kubota-Leopoldt $p$-adic zeta function implementation in SageMath?
Exactly as in the title. I am learning the $\mathbb{Z}_p^\times$ measure-theoretic construction of $p$-adic $L$-functions and was wondering if there was an `easy' way produce some example computations ...
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$L$-function of elliptic curves expansion into Dirichlet series
Let $E/\mathbb{Q}$ be an elliptic curve. The $L$-function of $E$ is defined to be the Euler product
$$
L_E(s) = \prod_{\text{ bad }p} (1 - a_p p^{-s})^{-1} \prod_{\text{ good }p} (1 - a_p p^{-s} + p^{...
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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\...
- 31
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Counting function and its Artin L-function
I am working with the following function: given a polynomial, $P(x)$, with non-negative integer coefficients and passing through origin, we can define
$$f(d)=\{1\leq a\leq d \ | \ P(a)\equiv 0 \mod(d)\...
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L-Function of Elliptic Curve and Modular Form
I have a problem to see how the galois-theoretic definition of the L-function of an elliptic curve gives the right answer and also the connection between the L-function of a weight 2 Hecke eigenform ...
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Why is it necessary to find extreme values of L-functions?
Apparently I have been exploring Riemann Zeta Function and have lately come across $L$-functions. After going through few papers, I realised that many mathematicians are giving their full time knowing ...
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References for this function: $\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}?$
Consider $$\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}$$
where $\chi$ is a Dirichlet character.
Is anything known about it?
I looked around and didn't see any references involving this ...
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Root number of the $L$-function of $y^2 = x^3 - n^2x$ and $n \pmod 8$.
Root number definition. Let $E_n$ be the elliptic curve $y^2 = x^3 - n^2 x$ where $n$ is a positive squarefree integer. It is known that the $L$-function of $E_n$, denoted $L(E_n,s)$, can be extended ...
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An $L-$function and a $J-$function. Related?
Consider a Dirichlet series for a non real character of modulus $q$
$$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$
and $s\in\Bbb C$ with real part greater than one.
Consider a $J$-series $$ J(s,...
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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
$$
L_p(E,s) = \begin{cases}(1-a_pp^...
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Limits with zeroes of L-functions
As usual, let $L(s,\chi) = \sum_{n\geq 1} \chi(n)n^{-s}$ be the L-function of a Dirichlet character mod $q$ and assume GRH. Since no nonzero meromorphic function has infinitely many zeroes in a ...
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Special values of Hecke $L$-function on imaginary quadratic fields
Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Let $\chi$ be an algebraic Hecke character on $K$ with conductor $\mathfrak{f}$ and infinity type $(a,b)$, i.e.
...
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Question on paper of Mazur, Tate, Teitelbaum and $p$-adic L functions of modular forms
I'm trying to fill in the details in proposition 14 of this paper by Mazur, Tate, and Teitelbaum. In particular, I'd like to understand the following.
Let $f$ be a cuspidal eigenform of weight $k$ and ...
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Euler Factors in Permutation Representation of Galois Group
Let $k$ be a number field and $K / \mathbb Q$ a Galois extension containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\operatorname{Gal}(K/k)$. Let $\chi$ denote the ...
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Order of vanishing of $L$ function
In the introduction of this paper, the authors say that: Let $M \in \mathbb{Z}_{>0}$. If $f$ is a normalised newform for $\Gamma_0(M)$ then we define
$$\Lambda(s,f)=(2\pi)^{-s}\Gamma(s) M^{s/2}L(s,...
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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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Real parts of non-trivial zeros of L-functions
Let $L_{\pi}$ be the L-function associated to an automorphic representation $\pi$ of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$, and let $R_{L_{\pi}}$ denote $\{\Re(s)-\frac{1}{2}|L_{\pi}(s)=0\wedge 0\...
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poles of local zeta integral in Tate's thesis
Let $F$ be a local field and $\omega : F^{\times} \to \mathbb {S}^1$ be a unitary character. The local zeta integral is defined to be
$$ z(s,\omega,f) = \int_{F^{\times}} f(x)\omega(x)\omega_s(x)d^{\...
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Required Background to Study L-Functions and Elliptic Curves
I am a mathematics student about to enter graduate school. I have interests in many areas of mathematics, but two areas of study that sound interesting are $L$-functions and elliptic curves. What ...
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Dirichlet $L$-function of primitive character in function field setting
Let $q=p^k$ be a prime power, and let $Q \in \mathbb{F}_q[t]$ be a polynomial. A Dirichlet character $\varphi$ of modulus $Q$ is a group homomorphism
$$
\varphi \colon (\mathbb{F}_q[t]/Q\mathbb{F}_q[t]...
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Moments of $L$-functions at real parts greater than 1
I am looking for a reference for the moments of $L$-functions evaluated at real parts greater than 1. I have looked and the only reference I can find are concerned with real parts between 0 and 1 (...
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Quadratic twist of modular L-function
(Sorry for my poor english..)
Let $F(z)=\sum_{n=1}^{\infty} a(n)q^n\in S_{2k}(\Gamma_0(N),\chi_0))$ be a newform with trivial character $\chi_0$. For $\text{Re}(s)>>0$, we can define
\begin{...
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Sum of the first $1/n$-th powers of z, L-function with rational powers.
The sum of the first $n$-th roots of some given complex number $z$ in the unit open disk, $|z|<1$
$$\sum_{n=1}^N z^{1/n},$$
could be expressed as a polynomial series on $N$ or $z$, or it might ...
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Non-vanishing of Dirichlet $L$-function $L(s,\chi)$ for $\Re(s)=1$ [duplicate]
I know that if $\chi$ is a non-principal Dirichlet character then the $L$-function $L(s,\chi)$ doesn't vanish for $s=1$. But, how about $s=1+it$ with $t\neq 0$? I found in this post: Zeros of ...
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Proof verification of non vanishing of $ ~L(1, \chi) \neq 0~$ for real valued character
I am self studying analytic number theory from Tom M Apostol introduction to analytic number theory and I am asking for solution verification for a part of Theorem 6.20 of Apostol.
I am adding it's ...
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