Does there exist a set $S \subset \mathbb N$ such that the Dirchlet density of $S$ is well-defined and positive, the Dirchlet density of $S \cap \operatorname{PRIMES}$ is well-defined and zero, and:

$ \prod_{n \in S} \frac{1}{1-n^{-s}}$

has a meromorphic continuation to the whole complex plane? Can we construct it?

Motivation: How special is the set of primes, among sets of natural numbers of approximately the same size, for having a meromorphic continuation? I would guess that analytic continuation should be a very rare occurrence, but I don't have an intuition for how rare, or how hard it is to find an example if one exists.

Extra credit for a functional equation.