Suppose that $S = \{\rho_{j} \}$ is any sequence of complex numbers such that $0 < \text{Re}(\rho_{j}) < 1$, $|\rho_{j}| \to \infty$ as j goes to infinity.
Then, can we construct a Dirichlet series $D(s) = \sum_{n \geq 1} a_{n}n^{-s}$ which converges absolutely for $\text{Re}(s) > 1$, analytic for $\text{Re}(s) > 0$, and \[ D(\rho_{j}) = 0 \] for all $j$?
If the given sequence $S$ is finite, then this is very easy; just put \[ D(s) = \prod_{j \leq N}(1 - A^{-s + \rho_{j}}) \] for some positive integer $A$, and this expands into some Dirichlet polynomial.
But due to some data, I think the infinite case can't be possible.
Is there any result on this topic?
Thank you all.