Dirichlet series
| L(s) = 1 | − 1.09·2-s − 0.714·3-s − 0.467·4-s − 0.147·5-s + 0.782·6-s + 0.374·7-s + 1.24·8-s − 0.645·9-s + 0.162·10-s + 0.498·11-s + 0.333·12-s − 0.635·13-s − 0.410·14-s + 0.105·15-s − 0.381·16-s + 1.81·17-s + 0.707·18-s − 0.705·19-s + 0.0691·20-s − 0.267·21-s − 0.546·22-s − 0.309·23-s − 0.886·24-s − 0.313·25-s + 0.695·26-s + 0.572·27-s − 0.174·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.8i) \, \Gamma_{\R}(s+5.12i) \, \Gamma_{\R}(s-21.8i) \, \Gamma_{\R}(s-5.12i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.04688\) |
| Root analytic conductor: | \(1.68425\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (21.8846650474i, 5.12584142776i, -21.8846650474i, -5.12584142776i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.25783743, −19.19598404, −17.68813753, −16.84293870, −14.28048914, −11.94606954, −10.04473558, −8.34438051, −0.69074565, 0.69074565, 8.34438051, 10.04473558, 11.94606954, 14.28048914, 16.84293870, 17.68813753, 19.19598404, 23.25783743