A converse theorem for Dirichlet <var>L</var>-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,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 (mod q), 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.