  This article consists to give a necessary and sufficient condition of the
meromorphic continuity of Dirichlet series defined as $\sum_{x\in \mathbf{N}^n}
\frac{a_{x}}{P(x)^s}$, Where $a_{x}$ is a $q$-automatic sequence of $n$
parameters and $P: \mathbf{C}^n \to \mathbf{C}$ a polynomial, such that $P$
does not have zeros on $\mathbf{Q}^{n}_{+}$. And some specific cases of $n=1$
will also be studied in this article as examples to show the possibility to
have an holomorphic continuity on the whole complex plane. Some equivalences
between infinite products are also built as consequences of these results.
