Dirichlet series
| L(s) = 1 | + (−0.105 − 0.750i)2-s + (1.23 + 0.0391i)3-s + (−0.447 − 0.592i)4-s + (0.134 − 0.154i)5-s + (−0.100 − 0.932i)6-s + (−0.900 + 0.477i)7-s + (0.0274 + 0.398i)8-s + (0.290 + 0.135i)9-s + (−0.130 − 0.0847i)10-s + (0.690 + 0.382i)11-s + (−0.529 − 0.750i)12-s + (−0.234 + 0.0235i)13-s + (0.453 + 0.626i)14-s + (0.172 − 0.185i)15-s + (−0.301 − 0.539i)16-s + (−0.506 + 0.533i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-14.1i) \, \Gamma_{\R}(s-2.38i) \, \Gamma_{\R}(s+16.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.22043\) |
| Root analytic conductor: | \(1.30460\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-14.141635588127452i, -2.380388488812225i, 16.522024076939676i:\ ),\ 1)\) |
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
−24.8976066984477, −22.4182283867368, −19.8925541329716, −13.6785015726197, −9.0428483882837, −7.2037747485489, −3.4008284908573, 9.3817470183602, 19.2459446523322, 20.2405029784980, 22.4944857591085, 24.9272215615215