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
−20.6092, −17.8923, −16.2656, −14.7021, −10.8635, −8.3003, −1.9512, 0.8682, 1.7167, 3.2655, 6.1016, 7.9119, 9.3511, 10.9515, 11.8637, 13.7645, 15.9090, 17.7636, 18.8677, 20.3705, 21.1413, 24.0335