Questions tagged [l-functions]

Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.

Filter by
Sorted by
Tagged with
5
votes
0answers
110 views

Non-vanishing of L-values for twists of Hilbert modular forms

Let $F$ be a real quadratic field, and let $f$ be a Hilbert modular form over $F$ of parallel weight 2. It's known, by a theorem of Rohrlich, that there exist infinitely many Hecke characters of $F$ ...
0
votes
1answer
176 views

On $L$-function of permutation representation

I came across the statement in a book: Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\...
0
votes
1answer
67 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
3
votes
1answer
117 views

English reference for the Brauer-Kuroda formula

I'm currently trying to understand the Brauer-Kuroda formula. Although there are many recent papers on the formula but they seem to be purely algebraic. They say that original analytic approach is ...
-2
votes
0answers
137 views

How to prove that $\mathrm{Aut}(\mathcal{M})\cong\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?

I want to study the structure of the rig of L-functions $\mathcal{M}$, which is defined as the maximal set of automorphic L-functions of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ for some $n$ that be ...
2
votes
0answers
91 views

The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
6
votes
2answers
400 views

Analytic equivalents for primes in arithmetic progressions

By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$. I would like ...
4
votes
0answers
106 views

Small values of $L(\frac{1}{2}, \chi_{-8d})$

In Section 3 of https://arxiv.org/pdf/0708.3990.pdf Soundararajan shows that there exists infinitely many fundamental discriminants $8d$ for which $L(1/2, \chi_{8d})$ is very small. His argument runs ...
1
vote
1answer
175 views

Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?

Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{...
3
votes
3answers
377 views

Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...
3
votes
0answers
187 views

Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?

To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$. Is the map $\pi\mapsto L_{\pi}$ bijective?
0
votes
0answers
108 views

Sets of L-functions being “almost bimonoids”

Let $\mathcal{M}$ be a set of L-functions (where by L-function I mean any L-function associated to an automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ which is an element ...
2
votes
0answers
61 views

Relations between spectral parameters of automorphic representations

Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated ...
6
votes
1answer
165 views

“Sub-logarithmic” zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
4
votes
1answer
204 views

Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?

Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
3
votes
0answers
106 views

Proving that central zero of an L-function is multiple

Central point of critical strip, ie. $s=1/2$, is conjectured to be the only argument at which an L-function can have multiple zero. This is interesting ia. for proving effective lower bounds for class ...
8
votes
1answer
308 views

Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion $$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$ and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
6
votes
1answer
367 views

Does the symmetric square L-function vanish at one?

Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one : Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
5
votes
1answer
283 views

Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
2
votes
2answers
220 views

Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$

I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475. Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ ...
4
votes
2answers
178 views

How to compute poles and values of L-functions?

Let $L(s)$ be an L-function, given by its series expansion and admitting an Euler product, say, for $s$ of large enough real part, $$L(s) = \sum_n \frac{a_n}{n^s} = \prod_p \prod_{i=1}^k (1-\alpha_i p^...
2
votes
0answers
225 views

Sum of inverses of zeros of $L$-functions

Let $L(s)=\sum_{n\ge1}a(n)/n^s$ be a "standard" $L$-function, say converging for $\Re(s)>k$, having an analytic continuation with no poles to an entire function of order $1$ with functional ...
12
votes
1answer
628 views

How do functional equations for zeta functions arise from the structure of a homology group?

I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this ...
2
votes
1answer
156 views

Modular symbols associated to Rankin Selberg convolutions and the symmetric square

I'm interested in understanding how one may associate modular symbols to the L-functions and $p$-adic L-functions associated to the Rankin Selberg convolution of two modular forms/ elliptic curves and ...
4
votes
0answers
55 views

Archimedean L-factors for symplectic group

Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...
4
votes
2answers
704 views

$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$

Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression $$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
4
votes
1answer
238 views

Effective, explicit bound on $L'(1,\chi)/L(1,\chi)$

Are there effective, explicit upper bounds on $L'(1,\chi)/L(1,\chi)$ in the literature, valid for all Dirichlet characters $\chi$? Due to the possibility of Siegel zeros, I don't imagine one can do ...
9
votes
1answer
231 views

Status of the extended form of the Lichtenbaum conjecture

The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$. ...
1
vote
0answers
145 views

Mixed Hodge structures over $F\otimes \mathbb{R}$

Let $F$ be a number field. Nekovàř, on page 18 of Values of L-functions and p-adic cohomology, is referring to the category of mixed Hodge structures over $F\otimes_{\mathbb{Q}} \mathbb{R}$. Can ...
2
votes
1answer
99 views

Calculation of Tate epsilon factor in the ramified case

Let $F$ be a nonarchimedean local field, $\chi$ a ramified character of $F^{\ast}$, $\psi$ a nontrivial character of $F$, and $dx$ a Haar measure on $F$ with respect to which the Fourier transform is ...
2
votes
1answer
213 views

Voronoi formula for the symmetric $L$-function with level $N $

Sorry to disturb. Does any experts here know something upon the Voronoi type for the symmetric $L$-functions$$\sum_{n\le X} A_F(1,n)e\left ( \frac{an}{c}\right)=?$$ Here $F$ is a symmetric-lift of a $...
15
votes
5answers
1k views

$|L'(1,\chi)/L(1,\chi)|$

Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$? Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\...
4
votes
0answers
145 views

Kronecker limit formula, modular curves, and the class number problem

Let $$Q(x,y)=ax^2+bxy+cy^2$$ be a positive definite quadratic form with $a>0$ and $D=b^2-4ac<0$. Let $$\zeta_Q(s)=\sideset{}{'}\sum_{m,n}Q(m,n)^{-s},$$ the accent indicating that $(0,0)$ is ...
11
votes
3answers
2k views

Are L-functions uniquely determined by their values at negative integers?

Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that the corresponding L-function $$L_{\{a_n\}...
6
votes
1answer
389 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
3
votes
2answers
236 views

Explicit formula: explicit work with general smoothing?

The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
3
votes
0answers
91 views

Similarity between two $L$-functions (Hasse-Weil $L$-function of twisted ellptic curve and Dirichlet $L$-function)

Let $E$ be an elliptic curve over $\mathbb Q$ with conductor $N$ and $E_d$ be its twisted curve by $d$, where $d$ is a fundamental discriminant with $(d,N)=1$. Let $\chi_d$ be a Dirichlet character ...
2
votes
1answer
144 views

Complex L-functions for Hermitian modular forms?

Fix an imaginary quadratic field $K$, and let $\mathcal{O}_K$ be its ring of integers. A Hermitian modular form of genus 1 (i.e., an automorphic form on $GU(1,1)$) of weight $(k_1,k_2)$ on a ...
4
votes
0answers
117 views

Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?

Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is ...
2
votes
0answers
106 views

Adelic Mellin transform with nontrivial character

This is a short question with a lot of setup. I apologize in advance. In Dan Bump's "Automorphic Forms and Representations," he constructs the L-function of a modular form via an "adelic Mellin ...
2
votes
0answers
116 views

Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$

This is a very short question. Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$. In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
10
votes
1answer
611 views

Least quadratic residue under GRH: an explicit bound

Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need ...
4
votes
2answers
192 views

Real non trivial zeros of Dirichlet L-functions

When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
4
votes
0answers
148 views

Second derivative at 1 of L function of elliptic curve

Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that $$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$ where $\gamma$ is Euler's ...
5
votes
2answers
249 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. 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 the ...
7
votes
0answers
251 views

A mysterious number related to Hasse-Weil L-function of elliptic curve

Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
4
votes
0answers
139 views

How many exceptional conductors are there?

We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
7
votes
1answer
308 views

Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
2
votes
0answers
94 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
4
votes
1answer
306 views

Modular forms and Period Polynomials

1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is ...

1
2 3 4 5
7