On the convolution of Hecke L ‐functions

Abstract

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grossencharakter Xi in Ki and consider the L-functionassociated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined bywhereand a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.