Dirichlet series
| L(s) = 1 | + 1.60·2-s − 0.226·3-s + 1.09·4-s + 1.37·5-s − 0.362·6-s + 0.302·7-s + 1.00·8-s + 1.15·9-s + 2.20·10-s − 1.49·11-s − 0.247·12-s + 0.460·13-s + 0.484·14-s − 0.311·15-s + 1.56·16-s − 2.28·17-s + 1.84·18-s − 0.174·19-s + 1.50·20-s − 0.0683·21-s − 2.39·22-s + 0.477·23-s − 0.226·24-s − 0.139·25-s + 0.738·26-s − 0.735·27-s + 0.331·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.3i) \, \Gamma_{\R}(s+6.19i) \, \Gamma_{\R}(s-21.3i) \, \Gamma_{\R}(s-6.19i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(11.2202\) |
| Root analytic conductor: | \(1.83021\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (21.3616736812i, 6.19782902262i, -21.3616736812i, -6.19782902262i:\ ),\ 1)\) |
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
−24.87079474, −23.47981713, −21.60728438, −17.85737806, −15.57558815, −13.50845517, −13.17316764, −10.47964700, −4.68419765, −1.99067701, 1.99067701, 4.68419765, 10.47964700, 13.17316764, 13.50845517, 15.57558815, 17.85737806, 21.60728438, 23.47981713, 24.87079474