Dirichlet series
| $L(s,f)$ = 1 | + (−0.250 + 0.433i)2-s + (0.150 − 0.786i)3-s + (−0.125 − 0.216i)4-s + (1.15 + 0.148i)5-s + (0.303 + 0.261i)6-s + (−0.186 − 0.826i)7-s +(0.125)·8-s + (−0.746 − 1.02i)9-s + (−0.354 + 0.464i)10-s + (0.166 + 0.581i)11-s + (−0.189 + 0.0658i)12-s + (−0.265 − 0.00552i)13-s + (0.404 + 0.126i)14-s + (0.291 − 0.889i)15-s + (−0.0312 + 0.0541i)16-s + (−0.711 + 1.25i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-9.63i) \, \Gamma_{\R}(s-1.37i) \, \Gamma_{\R}(s+11.0i) \, L(s,f)\cr
=\mathstrut & (-0.5 - 0.866i)\, \Lambda(1-s,\overline{f})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(4\) = \(2^{2}\) |
| \( \varepsilon \) | = | $-0.5 - 0.866i$ |
| primitive | : | yes |
| self-dual | : | no |
| Selberg data | = | $(3,\ 4,\ (-9.63244453i, -1.3740602838i, 11.006504814i:\ ),\ -0.5 - 0.866i)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1}
\end{aligned}\]