Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

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 an arbitrary, finite-dimensional, complex representation $\chi$ $\pi_1(X)$ admit meromorphic continuation to $\mathbb{C}$. Moreover, study behaviour function at $s=0$ point, it has zero order $\dim(\chi)(2g-2)$.