A comparison of zeros of \(L\)-functions
Let \(L(s) = L_1(s)/L_2(s)\) be the quotient of two automorphic \(L\)-functions for some \(\text{GL}(n)\) over a number field. The author proves in several cases that \(L(s)\) has infinitely many non-trivial poles. For the quotient of two Dirichlet \(L\)-functions for \({\mathbb Q}\) (automorphic \(L\)-functions for \(\text{GL}(1)\)) a proof is given by a simple trick using the converse theorem for \(\text{GL}(1)\) over \({\mathbb Q}\). Other examples use an extension of Hecke's converse theorem for \(\text{GL}(2)\) over \({\mathbb Q}\) proved by the author. The knowledge of analytic properties of the automorphic \(L\)-functions in question is necessary for application of the method.
scientific article; zbMATH DE number 1489492
scientific article