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Let $X$ be a compact, hyperbolic surface of genus $g\geq 2$. In this paper,we prove that the twisted Selberg and Ruelle zeta functions, associated with anarbitrary, finite-dimensional, complex representation $\chi$ of $\pi_1(X)$admit a meromorphic continuation to $\mathbb{C}$. Moreover, we study thebehaviour of the twisted Ruelle zeta function at $s=0$ and prove that at thispoint, it has a zero of order $\dim(\chi)(2g-2)$.

Twisted Ruelle zeta function at zero for compact hyperbolic surfaces