Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
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questions
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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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15 views
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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24 views
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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1answer
41 views
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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81 views
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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1answer
22 views
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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1answer
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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1answer
41 views
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
$$ z(s,\omega,f) = \int_{F^{\times}} f(x)\omega(x)\omega_s(x)d^{\...
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1answer
46 views
Required Background to Study L-Functions and Elliptic Curves
I am a mathematics student about to enter graduate school. I have interests in many areas of mathematics, but two areas of study that sound interesting are $L$-functions and elliptic curves. What ...
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1answer
42 views
Dirichlet $L$-function of primitive character in function field setting
Let $q=p^k$ be a prime power, and let $Q \in \mathbb{F}_q[t]$ be a polynomial. A Dirichlet character $\varphi$ of modulus $Q$ is a group homomorphism
$$
\varphi \colon (\mathbb{F}_q[t]/Q\mathbb{F}_q[t]...
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1answer
20 views
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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33 views
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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31 views
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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50 views
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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1answer
91 views
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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1answer
55 views
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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1answer
62 views
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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45 views
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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1answer
41 views
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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24 views
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, 447ā...
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1answer
61 views
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
$$L(f,s)=\prod_{p}\frac{1}{1 āc(p)p^{ās} + p^{kā1ā2s}}.$$
I wonder if/want to show that $f$ is actually a Hecke ...
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27 views
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
$$
F_d(s)=\sum_{n=1}^{\infty} \frac{a(d_0 n^2)}{(d_0 n^2)^s} \prod_{p|4Nnd}\left(1+\frac{1}{p}\right)^...
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29 views
Absolute value of roots of bad Euler factors.
Suppose $V$ is a smooth projective variety over number field $\mathbb{Q}$. For every prime integer $p$ and positive integer $j$, let
\begin{equation}
Z_p(H^j(V), T):=\det(1-{\rm Frob}_p\cdot T|H^j_{et}...
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1answer
83 views
Is the reciprocal of a Satake parameter a Satake parameter?
Let $F$ be an automorphic L-function for $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. Is it known whether for all but finitely many primes $p$ the set of Satake parameters of $F$ at $p$ is ...
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0answers
32 views
Do you know about the textbook of Selberg class of Dirichlet series?
I have read the Atle Selberg's thesis named "Old and new conjectures and results about a class of Dirichlet series". At the end of this thesis, he wrote
"A more complete account with proofs is ...
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26 views
Probability for an L-function to be RS-primitive
Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
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1answer
80 views
Minimum and maximum of a partial Euler product?
Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be
$$
\beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s};
$$
which for ...
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62 views
$p$- adic $L$- Function of a Dirichlet character at positive integers.
Let $p$ be a prime number and $d$ a natural number with $p\nmid d$. Furthermore let $\chi$ be a Dirichlet character with conductor $d$ and $G=Gal(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$.
In the ...
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162 views
Relationship between $\theta$ functions and number fields.
I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to.
Firstly, I know that $\theta$-functions arise as sections of ...
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1answer
73 views
Gamma Function Integral Identity
On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function:
$$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
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1answer
26 views
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 ...
3
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1answer
78 views
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 $k$ ...
3
votes
1answer
107 views
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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37 views
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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2answers
54 views
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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2answers
87 views
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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56 views
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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0answers
45 views
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 ...
4
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163 views
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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1answer
43 views
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 ...
2
votes
2answers
355 views
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 ...
5
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1answer
57 views
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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33 views
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 ...
1
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1answer
36 views
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$ ...
4
votes
1answer
87 views
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 $ \...
0
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1answer
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 ...
4
votes
1answer
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 ...
