Let $s=α+iβ$ be a complex number. Consider the Dirichlet series of the form $$f(s)=∑_{n=1}^{∞}(a_{n})/n^{s}$$ where $(a_{n})_{n≥1}$ is a real sequence.

We consider the class of Dirichlet series satisfying the following conditions:

(1) The real sequence $(a_{n})_{n≥1}$ is not identically zero.

(2) The function $f(s)$ is analytic for all $s=α+iβ∈ℂ$ such that $α>0$.

(3) The first term of $f(s)$ is not zero, i.e., $a₁≠0$.

(4) There exist a complex zero $a=1/2+iβ₁$ of $f $ in the critical strip: $s=α+iβ∈ℂ$ with $0<α<1$.

(5) $f $ has infinitely many non trivial zeros.

(6) For all $α∈(0,1)$ with $α≠1/2$ we have $f(α+iβ₁)≠0$.

**My question is**: Does there exist a known Dirichlet series verifying all these conditions and have non trivial zeros off the critical line.