Dirichlet series
| $L(s,f)$ = 1 | + 1.54·2-s + 0.246·3-s + 1.40·4-s + 0.737·5-s + 0.382·6-s − 0.261·7-s + 0.620·8-s − 0.939·9-s + 1.14·10-s − 0.953·11-s + 0.345·12-s + 0.278·13-s − 0.405·14-s + 0.181·15-s − 0.439·16-s + 1.30·17-s − 1.45·18-s + 0.0925·19-s + 1.03·20-s − 0.0645·21-s − 1.47·22-s + 1.13·23-s + 0.153·24-s − 0.456·25-s + 0.431·26-s − 0.478·27-s − 0.366·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+13.7i) \, \Gamma_{\R}(s-13.7i) \, L(s,f)\cr
=\mathstrut & \,\Lambda(1-s,f)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 1,\ (13.7797513519i, -13.7797513519i:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}
\end{aligned}\]