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
−24.915810, −22.365493, −20.847349, −17.035571, −15.316804, −12.277778, −10.510768, −9.030178, 6.535784, 7.150648, 9.323828, 11.663928, 13.455939, 15.064627, 16.793458, 18.443401, 19.252675, 24.180181, 24.954738