Dirichlet series
| $L(s,f)$ = 1 | − 0.707·2-s − 0.252·3-s + 0.5·4-s + 0.306·5-s + 0.178·6-s + 1.57·7-s − 0.353·8-s − 0.936·9-s − 0.216·10-s − 0.301·11-s − 0.126·12-s + 0.463·13-s − 1.11·14-s − 0.0772·15-s + 0.250·16-s − 0.759·17-s + 0.662·18-s + 1.62·19-s + 0.153·20-s − 0.397·21-s + 0.213·22-s + 0.419·23-s + 0.0891·24-s − 0.906·25-s − 0.327·26-s + 0.488·27-s + 0.787·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 22 ^{s/2} \, \Gamma_{\R}(s+1.72i) \, \Gamma_{\R}(s-1.72i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(22\) = \(2 \cdot 11\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 22,\ (1.72969651767i, -1.72969651767i:\ ),\ 1)\) |
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