  In this paper, we establish a simple criterion for two $L$-functions $L_1$
and $L_2$ satisfying a functional equation (and some natural assumptions) to
have infinitely many distinct zeros. Some related questions have already been
answered in the particular case of Automorphic forms using so-called Converse
Theorems. Deeper results can also be stated for elements of the Selberg class.
However, we shall give here a general answer that do not use any advanced
topics in analytic number theory. Therefore, this paper should be accessible to
anyone who has some basic notions in measure-theory and advanced complex
analysis.
