Dirichlet series
| L(s) = 1 | + (0.782 + 0.221i)2-s + (2.12 + 0.0498i)3-s + (−0.219 + 0.568i)4-s + (−0.256 + 0.842i)5-s + (1.65 + 0.510i)6-s + (1.17 + 0.298i)7-s + (0.0402 + 0.396i)8-s + (2.38 + 0.261i)9-s + (−0.387 + 0.602i)10-s + (0.338 − 0.0480i)11-s + (−0.493 + 1.19i)12-s + (−0.250 − 0.847i)13-s + (0.851 + 0.493i)14-s + (−0.586 + 1.77i)15-s + (0.771 + 0.0473i)16-s + (−0.840 + 0.214i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+26.8i) \, \Gamma_{\R}(s+13.4i) \, \Gamma_{\R}(s-40.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(58.7614\) |
| Root analytic conductor: | \(3.88774\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (26.84931332i, 13.468929556i, -40.31824288i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.315659, −20.875906, −19.591136, −14.390655, −9.111300, −7.860710, −4.624587, −3.911640, −2.190757, −1.177438, 1.825649, 2.947696, 3.741350, 4.981659, 7.368760, 8.102495, 9.103830, 10.957679, 12.933189, 13.969400, 14.640806, 15.297776, 17.673960, 19.170167, 20.180968, 21.389856, 22.476972, 24.248689