Dirichlet series
| L(s) = 1 | − 0.316·2-s + 0.0529·3-s − 1.15·4-s + 0.0261·5-s − 0.0167·6-s + 0.952·7-s + 0.447·8-s + 0.776·9-s − 0.00826·10-s + 0.818·11-s − 0.0612·12-s + 0.371·13-s − 0.301·14-s + 0.00138·15-s + 0.412·16-s + 0.332·17-s − 0.245·18-s − 0.259·19-s − 0.0301·20-s + 0.0504·21-s − 0.258·22-s + 0.686·23-s + 0.0236·24-s + 0.0289·25-s − 0.117·26-s + 0.135·27-s − 1.10·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+14.0i) \, \Gamma_{\R}(s+8.83i) \, \Gamma_{\R}(s-14.0i) \, \Gamma_{\R}(s-8.83i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.92296\) |
| Root analytic conductor: | \(1.77484\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (14.09372017438i, 8.83036460232i, -14.09372017438i, -8.83036460232i:\ ),\ 1)\) |
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
−23.790756277, −22.438056239, −21.050685750, −18.920976873, −17.571413650, −4.415386864, −1.115121568, 1.115121568, 4.415386864, 17.571413650, 18.920976873, 21.050685750, 22.438056239, 23.790756277