Dirichlet series
| L(s) = 1 | + (−0.0231 + 0.218i)2-s + (0.725 + 0.511i)3-s + (−1.24 − 0.0101i)4-s + (0.459 + 0.798i)5-s + (−0.128 + 0.146i)6-s + (1.08 − 0.509i)7-s + (0.0356 − 0.751i)8-s + (−0.547 + 0.742i)9-s + (−0.184 + 0.0819i)10-s + (0.254 + 0.0145i)11-s + (−0.898 − 0.644i)12-s + (0.0585 − 0.494i)13-s + (0.0862 + 0.247i)14-s + (−0.0749 + 0.814i)15-s + (0.704 + 0.0373i)16-s + (0.117 + 0.897i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.0i) \, \Gamma_{\R}(s+1.62i) \, \Gamma_{\R}(s-4.25i) \, \Gamma_{\R}(s-19.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.83941\) |
| Root analytic conductor: | \(1.16458\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.0947457086i, 1.628025115328i, -4.25436733726i, -19.4684034867i:\ ),\ 1)\) |
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
−24.00378304, −20.46928041, −18.43867388, −17.31901118, −14.43542522, −13.61952832, −11.93300114, −9.12479214, −8.42792671, −5.04136528, 8.46570807, 10.34402271, 13.74282677, 14.67734871, 17.60702206, 21.68550847, 23.07879845