Dirichlet series
| L(s) = 1 | − 0.784·2-s + 1.53·3-s + 0.411·4-s − 0.573·5-s − 1.20·6-s + 1.44·7-s − 0.946·8-s + 0.354·9-s + 0.450·10-s − 0.366·11-s + 0.631·12-s + 0.953·13-s − 1.13·14-s − 0.881·15-s + 0.274·16-s − 0.621·17-s − 0.277·18-s + 0.635·19-s − 0.235·20-s + 2.22·21-s + 0.287·22-s − 0.420·23-s − 1.45·24-s + 0.508·25-s − 0.747·26-s − 0.997·27-s + 0.595·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.0i) \, \Gamma_{\R}(s+2.71i) \, \Gamma_{\R}(s-20.0i) \, \Gamma_{\R}(s-2.71i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.86477\) |
| Root analytic conductor: | \(1.16857\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (20.0151956992i, 2.71117181968i, -20.0151956992i, -2.71117181968i:\ ),\ 1)\) |
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
−23.96854824, −20.63130728, −18.10349500, −15.37934645, −14.08391186, −11.35882929, −8.83893170, −8.06923257, 8.06923257, 8.83893170, 11.35882929, 14.08391186, 15.37934645, 18.10349500, 20.63130728, 23.96854824