A converse theorem for Dirichlet $L$-functions

Abstract

It is known that the space of solutions (in a suitable class of Dirichlet series with continuation over $C$) of the functional equation of a Dirichlet $L$-function $L(s,\chi )$ has dimension $\ge 2$ as soon as the conductor $q$ of $\chi$ is at least 4. Hence the Dirichlet $L$-functions are not characterized by their functional equation for $q\ge 4$. Here we characterize the conductors q such that for every primitive character $\chi (\mathrm{mod}q)$, $L(s,\chi )$ is the only solution with an Euler product in the above space. It turns out that such conductors are of the form $q=2a3bm$ with any square-free $m$ coprime to 6 and finitely many $a$ and $b$.