Let $G$ be a group with a finite symmetric set $S$ of generators.
Let $\ell_S(x)$ denote the word-length of a given $x\in G$.
For $s\in\mathbb C$ set
$$
Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s},
$$
where $G^*=G\smallsetminus\{1\}$.
It may happen that this sum converges for some $s$. 
It does not converge for free groups but does converge for abelian groups.
In the few simple cases I have computed, it turned out to be a linear combination of Riemann zeta functions with shifted arguments.

My question is a reference request: Has this kind of Dirichlet series been investigated? If so, I would like to have references.

Thank you.