Questions tagged [l-functions]

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

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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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Dirac Measure and Iwasawa Algebras

I am reading through some lecture notes on p-adic L-functions, and one of the exercises asks for $a \in \mathbb{Z}_p$ that we define the Dirac measure $\delta_a$ by $\int_{\mathbb{Z}_p} \phi \cdot \...
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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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Proof a 'well-known' result of Shimura on periods of modular forms

It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic....
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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 $k$ ...
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49 views

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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46 views

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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42 views

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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Prove continuity of L-function

I have just started learning about $L$-functions. The paper 'A History of the Prime Number Theorem' by Goldstein states that Dirichlet $L$-function $L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ ...
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Clarification in the definition of Critical strip for L functions (in particular of modular cusp forms and may be zeta function )

Let $f$ be a cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$. Define its associated L-series, $L(f,s)$ by $\sum_{n=1}^{\infty} \frac{a_f(n)}{n^s}$. One knows by Ramanujan-Petersson theorem(...
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Holomorphicity of the ratio of zeta functions

It is known that for a soluble extension of algebraic number fields $L:K$, the ratio of the Dedekind $\zeta$-functions $\frac{\zeta_L(s)}{\zeta_K(s)}$ is an entire function. It is also conjectured (...
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Question (Potentially Silly) about L functions

So I have never taken a class on Number Theory where L-functions would be discussed and I am learning about some things about L-functions by my own. Say $\chi$ denote any character of the group of ...
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Upper bound of a summation

Cross-post: https://mathoverflow.net/q/344996/123157 I'm studying with a paper 'Murty, M. Ram; Murty, V. Kumar Mean values of derivatives of modular L-series. Ann. of Math. (2) 133 (1991), no. 3, 447–...
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Euler product of the $L$-function implies Heck eigenform

Suppose $f(z)=\sum c(n)q^n$ and $L(f,s)=\sum c(n)n^{-s}$ have the Euler product $$L(f,s)=\prod_{p}\frac{1}{1 −c(p)p^{−s} + p^{k−1−2s}}.$$ I wonder if/want to show that $f$ is actually a Hecke ...
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Estimation of certain Euler product using a symmetric square $L$-function

Let $$ L(s)=\sum_{n=1}^{\infty}a(n)n^{-s} $$ be a modular $L$-function of conductor $N$, and let $$ F_d(s)=\sum_{n=1}^{\infty} \frac{a(d_0 n^2)}{(d_0 n^2)^s} \prod_{p|4Nnd}\left(1+\frac{1}{p}\right)^...
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Absolute value of roots of bad Euler factors.

Suppose $V$ is a smooth projective variety over number field $\mathbb{Q}$. For every prime integer $p$ and positive integer $j$, let \begin{equation} Z_p(H^j(V), T):=\det(1-{\rm Frob}_p\cdot T|H^j_{et}...
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Is the reciprocal of a Satake parameter a Satake parameter?

Let $F$ be an automorphic L-function for $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. Is it known whether for all but finitely many primes $p$ the set of Satake parameters of $F$ at $p$ is ...
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Do you know about the textbook of Selberg class of Dirichlet series?

I have read the Atle Selberg's thesis named "Old and new conjectures and results about a class of Dirichlet series". At the end of this thesis, he wrote "A more complete account with proofs is ...
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Probability for an L-function to be RS-primitive

Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
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Minimum and maximum of a partial Euler product?

Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be $$ \beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}; $$ which for ...
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$p$- adic $L$- Function of a Dirichlet character at positive integers.

Let $p$ be a prime number and $d$ a natural number with $p\nmid d$. Furthermore let $\chi$ be a Dirichlet character with conductor $d$ and $G=Gal(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$. In the ...
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Relationship between $\theta$ functions and number fields.

I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to. Firstly, I know that $\theta$-functions arise as sections of ...
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Gamma Function Integral Identity

On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function: $$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
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Expression of the dirichlet coefficient of an L-series in terms of the Satake parameters

Last year, I had found a pdf where the expression of the Dirichlet coefficient $\lambda_{\pi}(p^{\nu})$ in terms of the Satake parameters $\alpha_{p,i}(\pi)$ was given. Unfortunately I don't remember ...
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Asymptotics for average of Fourier coefficients of cusp form

Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function $$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$ of a weight $k$ ...
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Proof of Chebotarev Density Theorem without Artin Reciprocity

I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity. As I understand it, the main ...
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Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
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Need reference for proof of functional equation for generalized L-functions

When proving functional equation for Riemann zeta function one starts at the definition of gamma function $$\Gamma(s) = \int_0^{\infty} x^{s-1} e^x\mathrm dx\tag1$$ After a few steps we arrive at $$ ...
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Test vector for local zeta integral with ramified character

Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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Langlands L functions for groups over finite fields.

In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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Determining elliptic curve analytic rank even/odd

For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if ...
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Permutation group of Satake parameters

Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $ L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $ G_{p}$ of the ...
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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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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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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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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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113 views

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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152 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 ...