Dirichlet series
| L(s) = 1 | + (−0.925 + 0.988i)2-s + (0.212 + 0.453i)3-s + (0.805 − 0.841i)4-s + (0.361 + 0.218i)5-s + (−0.645 − 0.210i)6-s + (0.877 + 0.109i)7-s + (−0.747 + 1.57i)8-s + (−0.373 + 0.646i)9-s + (−0.549 + 0.154i)10-s + (−0.132 + 0.732i)11-s + (0.552 + 0.186i)12-s + (1.41 − 0.0366i)13-s + (−0.921 + 0.766i)14-s + (−0.0224 + 0.210i)15-s + (−0.213 − 1.19i)16-s + (0.268 + 0.640i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-24.6i) \, \Gamma_{\R}(s-6.06i) \, \Gamma_{\R}(s+30.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(18.4546\) |
| Root analytic conductor: | \(2.64262\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-24.632048i, -6.062229324i, 30.69427732i:\ ),\ 1)\) |
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
−24.0335, −21.1413, −20.3705, −18.8677, −17.7636, −15.9090, −13.7645, −11.8637, −10.9515, −9.3511, −7.9119, −6.1016, −3.2655, −1.7167, −0.8682, 1.9512, 8.3003, 10.8635, 14.7021, 16.2656, 17.8923, 20.6092