  Answering a question left open in previous research, we study the enumeration
of poles of the zeta function $\varphi(s)$ associated to an integer linear
recurrence sequence $\{a_n\}$. This enumeration can count poles more than once,
and we prove that this happens if and only if the module of relations of the
roots of the recurrence is nontrivial. A review of the existing literature on
the module of relations yields a series of sufficient conditions for the
enumeration of poles of $\varphi(s)$ to be injective. All of this is
illustrated by examples of both cases.
