Dirichlet series
| $L(s,f)$ = 1 | − 0.707·2-s + 1.45·3-s + 0.5·4-s − 1.20·5-s − 1.02·6-s − 0.439·7-s − 0.353·8-s + 1.10·9-s + 0.850·10-s + 0.301·11-s + 0.726·12-s + 0.206·13-s + 0.310·14-s − 1.74·15-s + 0.250·16-s + 0.459·17-s − 0.783·18-s + 0.139·19-s − 0.601·20-s − 0.637·21-s − 0.213·22-s + 1.47·23-s − 0.513·24-s + 0.447·25-s − 0.145·26-s + 0.157·27-s − 0.219·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 22 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.36i)) \, \Gamma_{\R}(s+(1 - 1.36i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(22\) = \(2 \cdot 11\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 22,\ (1 + 1.36941174922i, 1 - 1.36941174922i:\ ),\ 1)\) |
Euler product
\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)
Imaginary part of the first few zeros on the critical line