Determining \(L\)-functions by twists.
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.
scientific article; zbMATH DE number 2056170
scientific article