Functional Equations Characterize Dirichlet Characters

Abstract

We prove a converse theorem for functional equations of Dirichlet $L$-functions. Under mild assumptions, we prove that these functional equations for $L$-series of the form ${\sum }_{n\ge 1}f(n)n^{-s}$ force the coefficient function $f$ to be a primitive Dirichlet character. Consequently, these functional equations force the existence of an Euler product.