There exist weight 0 Maass cusp forms on Hecke congruence groups $\Gamma_0(N)$ with even character $\chi$. The L-function $L(s,f) = \sum a_n n^{-s}$ associated to the Maass cusp form $f$ has an Euler product of the form $L(s,f)= \prod_{p\mid N} \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N} \left(1 - a_p p^{-s} + \chi(p)p^{-2s} \right)^{-1}$ and satisfies a functional equation of the form $$\Lambda(s,f) = N^{s/2} \Gamma_{\R} \left(s + \delta + iR \right) \Gamma_{\R} \left(s + \delta - iR \right) \cdot L(s, f) = \varepsilon \Lambda(1-s,f),$$ where $N$ is the level, $R$ the eigenvalue of the Maass cusp form, $\varepsilon$ is the sign, and $\delta$ is 1 (or 0) when $f$ is odd (or even).

The dots in the plot correspond to L-functions with trivial character and $(R, N)$ as in the functional equation. The color shows whether the functional equation has sign +1 or sign -1. These have been found by a computer search. Click on any of the dots for details about the L-function.