  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. 
A Theorem for Distinct Zeros of L-Functions