Zeros of combinations of Euler products for $\sigma>1$

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.