Twisted Ruelle zeta function at zero for compact hyperbolic surfaces (Q2109397)

Let \(\Gamma\) be a discrete, torsion-free, cocompact subgroup of \(\mathrm{PSL}_2(\mathbb{R})\), \(X=\Gamma \backslash \mathbb{H}^2\) be a compact hyperbolic surface and let \(\chi:\Gamma \to \mathrm{GL}(V_\chi)\) be a finite-dimensional, complex representation of \(\Gamma\). In the present paper, it is proven that the twisted Selberg zeta function \(Z(s,\chi)\) admits a meromorphic continuation to \(\mathbb{C}\) and satisfies the functional equation \[ \frac{Z(s,\chi)}{Z(1-s,\chi)}=\exp \left \lbrack \dim(V_\chi) \mathrm{Vol}\int_0^{s-\frac{1}{2}} r\tan \pi r dr\right \rbrack, \] where the integral is a complex line integral along any curve from \(0\) to \(s-\frac{1}{2}\). Furthermore, The twisted Ruelle zeta function \(R(s, \chi)\) admits a meromorphic continuation to \(\mathbb{C}\) which satisfies the functional equation \[ R(s,\chi) R(-s,\chi)= (2\sin \pi s)^{2(2g-2)\dim V_\chi}. \] Therefore, The behavior of the twisted Ruelle zeta function \(R(s,\chi)\) near \(s = 0\) is given by \[ R(s,\chi)=\pm(2\pi s)^{\dim( V_\chi)(2g-2)}+ \text{ higher order terms}. \]