Dirichlet series
| $L(s,f)$ = 1 | +(0.5)·2-s + (1.06 − 0.481i)3-s +(0.25)·4-s + (−0.248 + 0.0721i)5-s + (0.533 − 0.240i)6-s + (0.676 + 0.291i)7-s +(0.125)·8-s + (−0.161 − 1.50i)9-s + (−0.124 + 0.0360i)10-s + (−0.264 + 0.294i)11-s + (0.266 − 0.120i)12-s + (−0.771 + 0.00362i)13-s + (0.338 + 0.145i)14-s + (−0.230 + 0.196i)15-s +(0.0625)·16-s + (0.335 + 0.143i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-8.23i) \, \Gamma_{\R}(s-2.64i) \, \Gamma_{\R}(s+10.8i) \, L(s,f)\cr
=\mathstrut & \, \Lambda(1-s,\overline{f})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(4\) = \(2^{2}\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | no |
| Selberg data | = | $(3,\ 4,\ (-8.239796i, -2.641226i, 10.881024i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1}
\end{aligned}\]