Dirichlet series
| $L(s,f)$ = 1 | + (0.556 − 0.928i)2-s + (−0.579 − 0.0446i)3-s + (0.179 − 1.03i)4-s + (0.652 + 0.534i)5-s + (−0.363 + 0.512i)6-s + (−0.493 + 0.0183i)7-s + (0.104 − 0.491i)8-s + (−0.572 + 0.0517i)9-s + (0.859 − 0.308i)10-s + (0.130 − 0.335i)11-s + (−0.149 + 0.590i)12-s + (0.259 + 0.340i)13-s + (−0.257 + 0.468i)14-s + (−0.354 − 0.338i)15-s + (−0.0961 − 1.12i)16-s + (0.282 + 0.246i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s-16.8i) \, \Gamma_{\R}(s-2.27i) \, \Gamma_{\R}(s+6.03i) \, \Gamma_{\R}(s+13.1i) \, L(s,f)\cr
=\mathstrut & \,\Lambda(1-s,\overline{f})
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | no |
| Selberg data | = | $(4,\ 1,\ (-16.89972715592i, -2.27258771492i, 6.03583588968i, 13.13647898116i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]