Dirichlet series
| $L(s,f)$ = 1 | + (−0.421 − 1.06i)2-s + (−0.768 + 1.31i)3-s + (−0.541 − 0.167i)4-s + (−0.400 − 0.239i)5-s + (1.72 + 0.266i)6-s + (−0.117 − 0.553i)7-s + (−0.268 + 0.648i)8-s + (−0.366 − 0.703i)9-s + (−0.0872 + 0.528i)10-s + (−0.0411 + 0.100i)11-s + (0.635 − 0.582i)12-s + (−0.309 + 0.328i)13-s + (−0.541 + 0.358i)14-s + (0.622 − 0.341i)15-s + (−0.0226 − 0.546i)16-s + (0.259 + 0.620i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s-16.4i) \, \Gamma_{\R}(s-0.171i) \, \Gamma_{\R}(s+16.5i) \, L(s,f)\cr
=\mathstrut & \,\Lambda(1-s,\overline{f})
\end{aligned}
\]
Invariants
| \( d \) | = | \(3\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | no |
| Selberg data | = | $(3,\ 1,\ (-16.403124740291375i, -0.17112189172831185i, 16.574246632019687i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1}
\end{aligned}\]