Dirichlet series
| L(s) = 1 | + (−0.549 − 0.530i)2-s + (−1.10 − 1.96i)3-s + (0.569 + 0.0527i)4-s + (−0.0706 + 0.631i)5-s + (−0.433 + 1.66i)6-s + (0.177 − 0.743i)7-s + (0.130 − 0.331i)8-s + (−1.51 + 2.38i)9-s + (0.373 − 0.309i)10-s + (0.485 + 0.0381i)11-s + (−0.527 − 1.17i)12-s + (0.584 + 0.471i)13-s + (−0.492 + 0.314i)14-s + (1.31 − 0.560i)15-s + (−0.456 − 0.691i)16-s + (−0.118 + 0.233i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-24.9i) \, \Gamma_{\R}(s-0.507i) \, \Gamma_{\R}(s+25.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.594774\) |
| Root analytic conductor: | \(0.840977\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-24.971575308i, -0.50792841772i, 25.479503724i:\ ),\ 1)\) |
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
−22.2442758, −20.6612096, −17.2730476, −16.2376972, −15.4281918, −11.6610295, −10.3465891, −8.6992935, −5.8752412, −4.3954280, 1.5766267, 6.4765890, 7.3912668, 10.8376046, 11.8005381, 13.5767069, 17.0047202, 18.3897038, 19.5433086, 23.2413714