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
−24.248689, −22.476972, −21.389856, −20.180968, −19.170167, −17.673960, −15.297776, −14.640806, −13.969400, −12.933189, −10.957679, −9.103830, −8.102495, −7.368760, −4.981659, −3.741350, −2.947696, −1.825649, 1.177438, 2.190757, 3.911640, 4.624587, 7.860710, 9.111300, 14.390655, 19.591136, 20.875906, 24.315659