Let $F(x) \in \mathbb{Z}[x]$ and $ \xi(s) = \sum^\infty_{n=1}g(n)n^{-s} $ be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series $ \xi_F(s) = \sum^\infty_{n=1}g(n)F(n)^{-s}. $ I call $\xi_F(s)$ the Dirichlet series obtained from $\xi(s)$ by shifting by $F(x)$. (Maybe this is known by another name.) I will start with a broad question: what can one say about such Dirichlet series? I would be greatful for any references.
Now I will be more specific.
1) Can one write down the coefficients of the Dirichlet series $\xi_F(s)$ (in terms of g(n) and F(n)) ?
2) It seems clear that if $\xi(s)$ converges for $\Re(s) > k$ then $\xi_F(s)$ converges for $\Re(s) > k/\deg(F)$. Suppose that $\xi(s)$ has meromorphic continuation to $\Re(s)> k+\epsilon$, for $\epsilon > 0$. Does it follow that $\xi_F(s)$ also has meromorphic continuation?
3) If 2) is true and the continued function $\xi(s)$ has a pole of order $d$ at $s=k$ does it follow that the continued function $\xi_F(s)$ has a pole of order $d$ at $s=k/\deg(F)$? or does it have a pole of some other order?
Perhaps, some of these question can be answered for special cases. For example, if $F(x) = x^p$ then $\xi_F(s)=\xi(ps)$ and all of the above is true.