Dirichlet series
| $L(s,f)$ = 1 | + 0.707·2-s − 0.380·3-s + 0.5·4-s − 0.253·5-s − 0.269·6-s + 0.397·7-s + 0.353·8-s − 0.855·9-s − 0.179·10-s + 1.20·11-s − 0.190·12-s − 0.948·13-s + 0.281·14-s + 0.0965·15-s + 0.250·16-s + 0.472·17-s − 0.604·18-s − 0.231·19-s − 0.126·20-s − 0.151·21-s + 0.849·22-s + 0.398·23-s − 0.134·24-s − 0.935·25-s − 0.670·26-s + 0.705·27-s + 0.198·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 2 ^{s/2} \, \Gamma_{\R}(s+(1 + 5.41i)) \, \Gamma_{\R}(s+(1 - 5.41i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(2\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | \((2,\ 2,\ (1 + 5.41733480684i, 1 - 5.41733480684i:\ ),\ 1)\) |
Euler product
\[\begin{aligned}L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\end{aligned}\]