Dirichlet series
| $L(s,f)$ = 1 | + (−0.250 − 0.433i)2-s + (0.360 + 0.356i)3-s + (−0.125 + 0.216i)4-s + (−0.592 + 0.583i)5-s + (0.0642 − 0.245i)6-s + (0.103 + 1.02i)7-s +(0.125)·8-s + (−0.357 + 0.613i)9-s + (0.400 + 0.110i)10-s + (1.34 + 0.122i)11-s + (−0.122 + 0.0334i)12-s + (0.0616 − 0.416i)13-s + (0.418 − 0.301i)14-s + (−0.421 − 0.000934i)15-s + (−0.0312 − 0.0541i)16-s + (−0.855 − 0.598i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-8.95i) \, \Gamma_{\R}(s-2.93i) \, \Gamma_{\R}(s+11.8i) \, L(s,f)\cr
=\mathstrut & (-0.5 + 0.866i)\, \Lambda(1-s,\overline{f})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(4\) = \(2^{2}\) |
| \( \varepsilon \) | = | $-0.5 + 0.866i$ |
| primitive | : | yes |
| self-dual | : | no |
| Selberg data | = | $(3,\ 4,\ (-8.9546625172i, -2.9365915306i, 11.8912540478i:\ ),\ -0.5 + 0.866i)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1}
\end{aligned}\]