Dirichlet series
| $L(s,f)$ = 1 | − 1.06·2-s − 0.456·3-s + 0.141·4-s − 0.290·5-s + 0.487·6-s − 0.744·7-s + 0.917·8-s − 0.791·9-s + 0.310·10-s + 0.166·11-s − 0.0644·12-s − 0.586·13-s + 0.795·14-s + 0.132·15-s − 1.12·16-s + 0.570·17-s + 0.845·18-s − 0.981·19-s − 0.0410·20-s + 0.339·21-s − 0.177·22-s + 0.662·23-s − 0.418·24-s − 0.915·25-s + 0.626·26-s + 0.817·27-s − 0.105·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+(1 + 9.53i)) \, \Gamma_{\R}(s+(1 - 9.53i)) \, L(s,f)\cr
=\mathstrut & -\,\Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $-1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 1,\ (1 + 9.53369526135i, 1 - 9.53369526135i:\ ),\ -1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}
\end{aligned}\]