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{equation} \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{equation} 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.

1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 Level R 9.53369526135 2.73410555925 4.13240421506 4.58236150313 5.10381155304 6.05402838077 6.37879720849 6.99327291741 7.78033646819 8.58191066592 9.23048002425 4.89723501573 6.73532969117 7.58531840144 8.01848237839 8.27366588959 9.02977428596 9.30018385212 9.83341273669 8.92287648699 7.22087197596 5.41733480684 5.09874190873 5.87935415776 8.04247759169 9.85989616224 3.70330780122 6.62042287384 8.52250301688 9.93491995937 4.13240421506 6.05402838077 6.82352699542 8.29469778357 8.4800158394 5.43618046142 7.98654905098 9.27729778527 9.39741755066 4.89723501573 6.35122321676 7.32552437933 8.2261459726 8.81789729862 3.02837629307 4.10322180991 5.70582652719 6.45847643848 7.58531840144 8.01848237839 8.03886120386 8.92287648699 9.65240704243 9.74374939916 3.84467228501 7.90114750377 8.77828239355 9.66046617807 6.87436376055 4.54845492142 6.12057553309 7.22087197596 8.66395307858 7.75813319502 8.27366588959 3.3088170303 9.37443036553 2.59237977171 6.75741527775 8.81974775988 9.29237932822 5.62812092942 7.22656906706 9.01322849206 4.64659164296 6.58769285315 9.4700831342 9.9039777038 6.22318082532 2.42505682709 6.32531406199 8.55288032905 7.48301811751 9.4700831342 9.9039777038 3.53600209294 5.50405667968 7.43179917172 8.03886120386 9.0800693497 9.74374939916 6.12057553309 7.75813319502 8.19303593168 8.77828239355 6.75741527775 9.29237932822 3.53600209294 5.50405667968 6.12057553309 6.64658135561 6.75741527775 7.43179917172 7.75813319502 9.08006934969 9.29237932822 8.03886120386 8.19303593168 8.77828239356 9.74374939916 8.69834295647 8.03886120386 9.74374939916 8.77828239355 6.12057553309 8.19303593168 4.38805356322 6.75741527775 7.75813319502 9.29237932822 6.64658135561 8.27366588959 4.82800766847 6.27191087352 7.33182919217 8.42175366822 8.69750590723 9.66612934925 3.45422650357 4.02170481942 5.58279102848 6.06885693385 7.35799566148 7.7010258369 8.15572563068 9.04317475934 9.25079724881 9.35463469751 1.92464430511 3.19085876294 4.71971646829 5.44273917126 5.82926875013 6.33448316434 7.05035422681 7.4314578005 8.07161868259 8.80251162268 8.94499430391 8.99915977043 9.98429702517 4.11900929292 5.10146081193 6.25758390694 7.05407006402 7.71956232873 8.25178553271 8.46236495549 9.20407719967 9.99016407004 8.82074088495 9.83341273669 6.00033540888 7.38627450582 6.30636358623 3.20437245526 6.4845713681 8.45261564535 9.78345165027 7.56348983462 5.42416298033 7.41262473822 9.95201540908 8.69834295647 8.21171118246 3.24997696354 7.69684780643 4.94783167713 9.19652685256 4.36801954023 7.74444773702 9.00120580965 9.81001812133 7.15749058532 5.8716820925 8.953063681 2.55636946085 4.18409504645 6.67376039246 7.788215329 9.44243727649 1.4463991334 4.64274919042 5.45975729468 7.34617187347 8.42338862095 9.46193719273 3.59362788512 6.01171674499