Dirichlet series
| $L(s,f)$ = 1 | + 0.289·2-s − 1.20·3-s − 0.916·4-s + 0.0395·5-s − 0.347·6-s + 0.448·7-s − 0.554·8-s + 0.444·9-s + 0.0114·10-s − 0.691·11-s + 1.10·12-s − 0.802·13-s + 0.129·14-s − 0.0475·15-s + 0.756·16-s − 1.03·17-s + 0.128·18-s + 0.637·19-s − 0.0362·20-s − 0.538·21-s − 0.200·22-s + 0.508·23-s + 0.666·24-s − 0.998·25-s − 0.232·26-s + 0.667·27-s − 0.410·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+(1 + 12.1i)) \, \Gamma_{\R}(s+(1 - 12.1i)) \, L(s,f)\cr
=\mathstrut & -\,\Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $-1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 1,\ (1 + 12.1730083247i, 1 - 12.1730083247i:\ ),\ -1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}
\end{aligned}\]