Dirichlet series
| L(s) = 1 | + (−0.476 + 0.267i)2-s + (2.01 + 0.0977i)3-s + (0.631 + 0.0125i)4-s + (−0.0909 − 0.0586i)5-s + (−0.986 + 0.493i)6-s + (1.19 − 0.0783i)7-s + (0.396 + 0.163i)8-s + (2.03 + 0.491i)9-s + (0.0590 + 0.00357i)10-s + (−0.723 − 1.15i)11-s + (1.27 + 0.0871i)12-s + (0.372 − 0.200i)13-s + (−0.550 + 0.358i)14-s + (−0.177 − 0.127i)15-s + (−0.411 + 0.471i)16-s + (−0.217 − 0.423i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-30.3i) \, \Gamma_{\R}(s-3.58i) \, \Gamma_{\R}(s+33.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(14.8407\) |
| Root analytic conductor: | \(2.45745\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-30.34363158i, -3.587613116i, 33.9312447i:\ ),\ 1)\) |
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
−23.97808, −21.07706, −20.25727, −19.41168, −17.84994, −15.45637, −14.58262, −13.23412, −11.10762, −9.51519, −8.20409, −7.37832, −4.31164, −2.43873, −1.76805, 1.92129, 7.93975, 8.42169, 10.73587, 13.62824, 14.72259, 16.29832, 18.69241, 20.13488, 21.01137, 24.44886