非線形問題と複素幾何学
《非線形問題と複素幾何学》 · 第 60 页 · Section 3.8「Hecke characters」，Theorem 3.8
Moreover, $\Lambda(s,\psi)$ satisfies the functional equation $\Lambda(s,\psi)=W(\psi)\Lambda(1-s,\overline\psi)$ with the root number $W(\psi)$ given by the corresponding Gauss sum $W(\psi)=\tau(\psi)(N\mathfrak m)^{-1/2}$.