  We study the distribution of the zeros of functions of the form $f(s)=h(s)
\pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line,
$a$ a real number. One of our results establishes sufficient conditions under
which all but finitely many of the zeros of $f(s)$ lie on the line $\Re s = a$,
called the {\it critical line} for the function $f(s)$, and be simple, given
that all but finitely many of the zeros of $h(s)$ lie on the half-plane $\Re s
< a$. This results can be regarded as a generalization of the necessary
condition of stability for the function $h(s)$, in the Hermite-Biehler theorem.
We apply this results to the study of translations of the Riemann Zeta Function
and $L$ functions, and integrals of Eisenstein Series, among others.
