Dirichlet series
| L(s) = 1 | + (−0.805 − 0.346i)2-s + (1.01 − 0.482i)3-s + (1.33 + 0.211i)4-s + (0.0135 + 0.737i)5-s + (−0.983 + 0.0379i)6-s + (0.493 − 0.322i)7-s + (−0.768 − 0.632i)8-s + (−0.219 − 1.46i)9-s + (0.244 − 0.598i)10-s + (0.736 + 0.176i)11-s + (1.45 − 0.429i)12-s + (1.35 − 0.218i)13-s + (−0.508 + 0.0885i)14-s + (0.369 + 0.740i)15-s + (0.742 + 0.137i)16-s + (−0.710 − 0.538i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+31.1i) \, \Gamma_{\R}(s+14.6i) \, \Gamma_{\R}(s-45.7i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(83.9458\) |
| Root analytic conductor: | \(4.37857\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (31.1037294i, 14.63912698i, -45.7428564i:\ ),\ 1)\) |
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
−24.74877, −20.92595, −19.66443, −16.52683, −10.96495, −8.72222, −8.10403, −6.02834, −4.00877, −2.26306, −1.49102, 0.67559, 1.75662, 2.68752, 3.70084, 6.41295, 7.06783, 8.37063, 9.37561, 10.96367, 11.54345, 13.38900, 14.67232, 15.54082, 17.32147, 18.36792, 19.59650, 20.19161, 21.17269, 23.19594, 24.62502