Let \(\chi\) be a character mod q, and form the Gaussian sum \[ G(m,\chi)=q^{-m^ 2/2}\sum_{D mod q}\chi (Det D) \exp (\frac{2\pi i}{q}Tr(D)), \] where m is an element of \({\mathbb{N}}\), the set of positive rational integers, D runs over all integral \(m\times m\) matrices mod q, and Tr(D) is the trace of D.
In his book [Selberg’s Zeta-, L-, and Eisenstein-series (Lect. Notes Math. 1030) (1983; Zbl 0519.10018)], the author proved Theorem 71: Let \(m\in {\mathbb{N}}\) and \(\chi\) be an even primitive character mod q. Then \(G(m,\chi)=(G(1,\chi))^ m.\) And he conjectured that the theorem would be valid for odd primitive characters.
In the present paper, he proves that (i) the above conjecture is true, and (ii) a generalization of Maass’ zeta-function, which is called Maass’ L-series in this paper, that is \[ \zeta (\chi,u,v,S,z)=\sum_{G}\frac{\chi (Det G_ 1) u(SG) w(G'S'SG)}{Det(G'S'SG)^ z},\quad G=\left( \begin{matrix} G_ 1\\ G_ 2\end{matrix} \right), \] can, for a primitive character \(\chi\), be analytically continued to the whole z-plane and satisfies a functional equation. For the notations not explained here, one may refer to the above Lecture Note or the original paper.