Well-done! I used Hadamard's factorization theorem in a lemma. I will look at some literature on the subject. Thanks again!

Many thanks for your answer and your references. I was more or less convinced that both of the corollaries were already known. Nevertheless, my result works for general Dirichlet series and, as far as I know, requires less (differents) assumptions than those of the Selberg class. Of course, my conclusions are weaker. Is my result still of interest?

I actually went to the fundamental mathematics' lab at my university this afternoon. Unfortunately (or not!), it's the holidays in France and they ends in two weeks. I can not wait! But I know that a direct answer will always be preferable.

If $f$ is a cusp form, then it vanishes at all cusps. Else, $f$ is a linear combinaison of Eisenstein series : thus, can we find a certain basis for the Eisenstein-space for which one knows the constant term at each cusps of each element of the basis ?

@GHfromMO : Yes that is almost what I want : but I am only interested in the constant term.

@PeterHumphries : I heard that the Atkin-Lehner involution method was only working in the case where $f$ is a cusp form and a new form. Can you please explain me the general method ? (Thanks !)