Dirichlet series
| L(s) = 1 | + (−0.438 − 0.674i)2-s + (0.132 + 0.801i)3-s + (0.175 − 0.0833i)4-s + (1.09 − 0.416i)5-s + (0.482 − 0.440i)6-s + (−0.109 − 0.676i)7-s + (0.219 − 0.0817i)8-s + (−0.756 + 1.01i)9-s + (−0.760 − 0.555i)10-s + (−0.928 − 1.62i)11-s + (0.0900 + 0.129i)12-s + (0.468 + 0.628i)13-s + (−0.408 + 0.370i)14-s + (0.479 + 0.821i)15-s + (−0.569 − 0.941i)16-s + (0.983 + 0.326i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+31.1i) \, \Gamma_{\R}(s+9.55i) \, \Gamma_{\R}(s-40.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(48.7085\) |
| Root analytic conductor: | \(3.65203\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (31.12026336i, 9.550875644i, -40.671139i:\ ),\ 1)\) |
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
−23.0149, −20.7115, −18.3235, −17.4003, −15.0847, −12.9097, −7.3198, −5.7941, −2.6968, −1.5135, 0.5315, 1.7934, 3.0568, 4.8013, 5.9492, 8.0871, 9.4353, 10.5424, 11.0261, 13.3839, 14.0789, 16.3218, 16.7194, 18.6741, 19.7888, 21.0413, 21.6572, 23.4876