Dirichlet series
| L(s) = 1 | − 1.22·2-s − 1.08·3-s − 0.509·4-s + 0.363·5-s + 1.32·6-s − 0.885·7-s + 1.86·8-s − 0.269·9-s − 0.446·10-s − 0.622·11-s + 0.552·12-s + 0.755·13-s + 1.08·14-s − 0.394·15-s − 0.758·16-s − 0.749·17-s + 0.330·18-s − 0.223·19-s − 0.185·20-s + 0.960·21-s + 0.762·22-s + 0.519·23-s − 2.02·24-s + 0.585·25-s − 0.925·26-s + 0.777·27-s + 0.450·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.2i) \, \Gamma_{\R}(s+5.61i) \, \Gamma_{\R}(s-19.2i) \, \Gamma_{\R}(s-5.61i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.44396\) |
| Root analytic conductor: | \(1.65177\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (19.22318167776i, 5.61130096696i, -19.22318167776i, -5.61130096696i:\ ),\ 1)\) |
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
−23.274395663, −22.373389050, −17.967891299, −16.649461689, −13.241374768, −10.662580235, −8.992144478, −0.501720138, 0.501720138, 8.992144478, 10.662580235, 13.241374768, 16.649461689, 17.967891299, 22.373389050, 23.274395663