On the theory of normalized Shintani \(L\)-functions and its application to Hecke \(L\)-functions. I. Real quadratic fields (Q669218)

A previous paper of the author and \textit{N. Sato} [Math. Z. 280, No. 3--4, 1085--1092 (2015; Zbl 1391.11104)] developed a theory of normalized multivariable Shintani \(L\)-functions. The main goal of the present paper is to prove a functional equation for multivariable generalizations of Hecke \(L\)-functions using Shintani's approach and then apply this theory to real quadratic fields. Given a Hecke \(L\)-function \(L_{\chi}(s)\) for a real quadratic field, an entire function \(F(s_1, s_2)\) is constructed such that \(F(s, s)\) is a slight modification of \(L_{\chi}(s)\) and \(F(s_1, s_2)\) satisfies a functional equation.