Functional equations for Selberg zeta functions with Tate motives (Q2239165)

For a compact Riemann surface \(M\) of genus \(g \geq 2\) and the Selberg zeta function \(Z_M(s)\), define the function \(Z_{M(f)}(s)\) with Tate motives \(f\) as \[ Z_{M(f)}(s)=\prod_k\zeta_M(s-k)^{a(k)} \] for a Laurent polynomial \[ f(x)=\sum_{k \in \mathbb{Z}}a(k)x^k \in \mathbb{Z}[x,x^{-1}]. \] The authors prove that the functional equations of the form \(\zeta_{M(f)}(1-s)=\zeta_{M(f)}(s)\) hold if and only if so called the absolute automorphy condition is satisfied, i.e., if \(f(x^{-1})=-x^{-D}f(x)\) for each integer \(D\).