Determining \(L\)-functions by twists.
type: Item
instance of: scholarly article
title: Determining \(L\)-functions by twists.
zbMATH Open document ID: 1060.11029
DOI: 10.1016/S0022-314X(03)00158-6
author: D. Massart
published in: Journal of Number Theory
publication date: 2004-03-14T00:00:00Z
review text: Suppose that \(L_1(s)\) and \(L_2(s)\) are two \(L\)-functions whose twists by a set of Dirichlet characters simultaneously vanish (or vanish modulo a set of primes in a suitable sense) at a critical point. The author examines the extent to which this property determines the \(L\)-functions in the cases of Dirichlet \(L\)-functions, \(L\)-functions of abelian varieties over finite fields, and \(L\)-functions associated to modular forms. The obtained interesting results are presented by seven theorems. As an example we state one of them. Suppose that \(f_i\in S_{2k}(M_i, \chi_0)\), \(i=1,2\), are normalized newforms of weight \(2k\), and trivial nebentypus. Let \(L(s, f_i)\) be their associated \(L\)-functions. Suppose that \[ L_{\chi}(k, f_1)=0\Leftrightarrow L_{\chi}(k, f_2)=0 \] for all quadratic Dirichlet characters \(\chi\). Then \(M_1\) and \(M_2\) are equal up to square factors. At the end of the paper the author poses two open problems.
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 2056170
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: twists
zbMATH Keywords: special values
reviewed by: Ramūnas Garunkštis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2056170