Dirichlet series
| L(s) = 1 | − 1.50·2-s − 0.487·3-s + 0.926·4-s + 0.169·5-s + 0.735·6-s − 0.471·7-s − 0.872·8-s − 1.72·9-s − 0.255·10-s − 1.17·11-s − 0.452·12-s + 0.0890·13-s + 0.711·14-s − 0.0826·15-s + 1.34·16-s − 1.57·17-s + 2.60·18-s − 0.153·19-s + 0.156·20-s + 0.230·21-s + 1.76·22-s − 2.27·23-s + 0.425·24-s + 1.40·25-s − 0.134·26-s + 1.31·27-s − 0.429·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.7i) \, \Gamma_{\R}(s+10.7i) \, \Gamma_{\R}(s-16.7i) \, \Gamma_{\R}(s-10.7i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(20.9720\) |
| Root analytic conductor: | \(2.13998\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.77928785768i, 10.78021484612i, -16.77928785768i, -10.78021484612i:\ ),\ 1)\) |
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
−24.089955178, −22.439762335, −20.157973750, −18.090841002, −8.590781821, −5.897261184, −2.591708809, −0.101811828, 0.101811828, 2.591708809, 5.897261184, 8.590781821, 18.090841002, 20.157973750, 22.439762335, 24.089955178