  In this paper we consider Dirichlet series absolutely converging for
$\sigma>1$ with an Euler product, natural bounds on the coefficients and
satisfying orthogonality relations of Selberg type. Let $N\geq 1$,
$F_1(s),...,F_N(s)$ be as above and $P(X_1,...,X_N)$ be a non-monomial
polynomial with coefficients in the ring of $p$-finite Dirichlet series
absolutely converging for $\sigma\geq 1$; then $P(F_1(s),\ldots,F_N(s))$ has
infinitely many zeros for $\sigma>1$. Our result in particular applies to Artin
$L$-functions, automorphic $L$-functions under the Ramanujan conjecture, and
the elements of the Selberg class with polynomial Euler product under the
Selberg orthonormality conjecture. This extends the work of Booker and Thorne,
who proved the same result for automorphic $L$-functions under the Ramanujan
conjecture. Our proof avoids to use the properties of twists by Dirichlet
characters, a key point in Booker and Thorne's proof, replacing them by results
on the Dirichlet density of non-zero coefficients of $L$-functions of the above
type.
