An evaluation of the central value of the automorphic scattering determinant

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $ϕ(s)$ denote the automorphic scattering determinant. From the known functional equation $ϕ(s)ϕ(1-s)=1$ one concludes that $ϕ(1/2)^{2} = 1$. However, except for the relatively few instances when $ϕ(s)$ is explicitly computable, one does not know $ϕ(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $ϕ(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $ϕ(s)$. Then $ϕ(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.

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