On the functional equation of the normalized Shintani l-function of several variables

TL;DR

This paper introduces the normalized Shintani L-function of multiple variables, establishes its functional equation, and explores its special values, generalizing known properties of the Hurwitz-Lerch zeta function.

Contribution

It presents the first integral representation and functional equation for the multi-variable normalized Shintani L-function, extending classical zeta function results.

Findings

Proved the functional equation of the normalized Shintani L-function.

Derived special values at non-positive and positive integers.

Generalized the functional equation of the Hurwitz-Lerch zeta function.

Abstract

In this paper, we introduce the normalized Shintani L-function of several variables by an integral representation and prove its functional equation. The Shintani L-function is a generalization to several variables of the Hurwitz-Lerch zeta function and the functional equation given in this paper is a generalization of the functional equation of Hurwitz-Lerch zeta function. In addition to the functional equation, we give special values of the normalized Shintani L-function at non-positive integers and some positive integers.