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L-functions of signature (0,0,0;)

These L-functions satisfy a functional equation with \(\Gamma\)-factors \[\begin{aligned}\Gamma_\R(s + i \mu_1)\Gamma_\R(s + i \mu_2)\Gamma_\R(s + i \mu_3)\end{aligned}\] with \(\mu_j\in \R\) and \(\mu_1 + \mu_2 + \mu_3 = 0\). By permuting and possibly taking the complex conjugate, we may assume \(\mu_1 \ge \mu_2 \ge 0\), so the functional equation can be represented by a point \( (\mu_1, \mu_2) \) below the diagonal in the first quadrant of the Cartesian plane.

L-functions of conductor 1

The dots in the plot correspond to L-functions with \((\mu_1,\mu_2)\) in the \(\Gamma\)-factors, colored according to the sign of the functional equation (blue indicates \(\epsilon=1\)). Click on any of the dots for detailed information about the L-function.