Functional Equations Characterize Dirichlet Characters

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.