On the theory of normalized Shintani L-function and its application to Hecke L-function

Abstract

We define the class of normalized Shintani L-functions of several variables. Unlike Shintani zeta functions, the normalized Shintani L-function is a holomorphic function. Moreover it satisfies a good functional equation. We show that any Hecke L-function of a totally real field can be expressed as a diagonal part of some normalized Shintani L-function of several variables. This gives a good several variables generalization of a Hecke L-function of a totally real field. This also gives a new proof of the functional equation of the Hecke L-function of a totally real field.