Dirichlet series
| L(s) = 1 | − 2.61·2-s − 0.290·3-s + 3.59·4-s + 1.40·5-s + 0.761·6-s − 0.305·7-s − 3.50·8-s + 0.634·9-s − 3.68·10-s + 1.66·11-s − 1.04·12-s − 0.554·13-s + 0.798·14-s − 0.409·15-s + 3.33·16-s − 1.17·17-s − 1.66·18-s − 0.0100·19-s + 5.05·20-s + 0.0888·21-s − 4.34·22-s + 1.38·23-s + 1.01·24-s − 0.207·25-s + 1.45·26-s − 0.635·27-s − 1.09·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.4i) \, \Gamma_{\R}(s+10.0i) \, \Gamma_{\R}(s-17.4i) \, \Gamma_{\R}(s-10.0i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(19.8698\) |
| Root analytic conductor: | \(2.11129\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.4955826325i, 10.06392007128i, -17.4955826325i, -10.06392007128i:\ ),\ 1)\) |
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
−24.80614873, −21.72346929, −19.44167920, −17.37454622, −9.30707659, −6.61355595, −1.87541403, −0.81810660, 0.81810660, 1.87541403, 6.61355595, 9.30707659, 17.37454622, 19.44167920, 21.72346929, 24.80614873