Functional equations of Selberg and Ruelle zeta functions for non-unitary twists

Abstract

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable s in some right-half plane of $${\mathbb {C}}$$. In Spilioti (Ann Glob Anal Geom 53(2):151–203, 2018), it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at s with those at $$-s$$. We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators.