  We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$
satisfies a linear recurrence of arbitrary degree with integer coefficients.
Under suitable hypotheses, we prove that it has a meromorphic continuation to
the complex plane, giving explicit formulas for its pole set and residues, as
well as for its finite values at negative integers, which are shown to be
rational numbers. To illustrate the results, we focus on some concrete examples
which have also been studied previously by other authors.
