# A converse theorem for Dirichlet $L$-functions

### Jerzy Kaczorowski

Adam Mickiewicz University, Poznan, Poland### Giuseppe Molteni

Università di Milano, Italy### Alberto Perelli

Università di Genova, Italy

## 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 $L(s,χ)$ has dimension $≥2$ as soon as the conductor $q$ of $χ$ is at least 4. Hence the Dirichlet $L$-functions are not characterized by their functional equation for $q≥4$. Here we characterize the conductors q such that for every primitive character $χ(modq)$, $L(s,χ)$ 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 $b$.

## Cite this article

Jerzy Kaczorowski, Giuseppe Molteni, Alberto Perelli, A converse theorem for Dirichlet $L$-functions. Comment. Math. Helv. 85 (2010), no. 2, pp. 463–483

DOI 10.4171/CMH/202