On a generalized Eulerian product defining a meromorphic function on the whole complex plane

The paper studies the Euler product of the form
$$ P_\pi(M,a(p)|\alpha)=\prod_{p\in P(M)}\left(1-\frac{a(p)}{p^{\alpha+\pi(p)}}\right)^{-1}, $$
where $M$ is an arbitrary monoid of natural numbers formed by the set of primes $P(M)$.
Another object of study is the Dirichlet series of the form
$$ f_\pi(M|\alpha)=\sum_{n\in M}\frac{1}{n^{\alpha +\pi(n)}}. $$

It turns out that they have completely different properties. The Dirichlet series $f_\pi (M| \alpha)$ defines a holomorphic function on the entire complex plane.
And the Euler product $P_\pi(M| \alpha)$ for a monoid $M$ whose set of primes $P(M)$ is infinite, sets on the entire complex plane a meromorphic function that has a countable set of special vertical lines, each of which has a countable set of poles.
In conclusion, the relevant problem of the zeros of the function $f_\pi(M|\alpha)$ is considered.