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
\[\begin{aligned}
\Lambda(s,f) = N^{s/2} \Gamma_{\R}
\left(s + \delta + iR \right) \Gamma_{\R}
\left(s + \delta - iR \right) \cdot L(s, f) =
\varepsilon \Lambda(1-s,f),
\end{aligned}\]
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.