Dirichlet series
| L(s) = 1 | + 1.72·2-s + 1.36·3-s + 0.489·4-s + 0.203·5-s + 2.33·6-s − 0.136·7-s − 1.68·8-s + 0.659·9-s + 0.350·10-s + 0.181·11-s + 0.665·12-s + 0.0790·13-s − 0.235·14-s + 0.277·15-s − 2.14·16-s + 0.585·17-s + 1.13·18-s − 0.106·19-s + 0.0997·20-s − 0.185·21-s + 0.312·22-s + 1.05·23-s − 2.29·24-s + 0.423·25-s + 0.135·26-s + 0.637·27-s − 0.0668·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+18.9i) \, \Gamma_{\R}(s+4.98i) \, \Gamma_{\R}(s-18.9i) \, \Gamma_{\R}(s-4.98i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.66914\) |
| Root analytic conductor: | \(1.54304\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (18.90093346964i, 4.98219315884i, -18.90093346964i, -4.98219315884i:\ ),\ 1)\) |
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
−23.626790960, −22.325094986, −20.949271050, −14.813509211, −13.944514504, −12.745827919, −8.998929026, −3.206090118, 3.206090118, 8.998929026, 12.745827919, 13.944514504, 14.813509211, 20.949271050, 22.325094986, 23.626790960