Dirichlet series
| L(s) = 1 | + (−0.513 − 0.833i)2-s + (−0.0450 + 1.33i)3-s + (0.0506 + 0.855i)4-s + (−0.0405 − 0.440i)5-s + (1.13 − 0.645i)6-s + (−0.317 + 0.0815i)7-s + (−0.0737 − 0.0491i)8-s + (−0.363 − 0.119i)9-s + (−0.345 + 0.259i)10-s + (0.986 + 0.239i)11-s + (−1.14 + 0.0289i)12-s + (−0.298 − 0.973i)13-s + (0.230 + 0.222i)14-s + (0.587 − 0.0341i)15-s + (−0.0210 + 0.498i)16-s + (−0.151 + 0.392i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.4i) \, \Gamma_{\R}(s+1.20i) \, \Gamma_{\R}(s-3.81i) \, \Gamma_{\R}(s-19.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18063\) |
| Root analytic conductor: | \(1.04238\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.49640718i, 1.201562854542i, -3.81917738186i, -19.8787926526i:\ ),\ 1)\) |
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
−24.954738, −24.180181, −19.252675, −18.443401, −16.793458, −15.064627, −13.455939, −11.663928, −9.323828, −7.150648, −6.535784, 9.030178, 10.510768, 12.277778, 15.316804, 17.035571, 20.847349, 22.365493, 24.915810