Dirichlet series
| L(s) = 1 | + (−0.712 + 1.53i)2-s + (1.50 + 0.279i)3-s + (−1.13 − 0.652i)4-s + (−0.0304 − 0.797i)5-s + (−1.49 + 2.10i)6-s + (−0.355 + 0.815i)7-s + (−0.0527 − 1.27i)8-s + (0.673 + 1.11i)9-s + (1.24 + 0.521i)10-s + (−0.294 − 1.03i)11-s + (−1.51 − 1.29i)12-s + (−0.224 − 0.782i)13-s + (−0.998 − 1.12i)14-s + (0.177 − 1.20i)15-s + (1.47 + 0.157i)16-s + (0.697 + 0.337i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.1i) \, \Gamma_{\R}(s+0.433i) \, \Gamma_{\R}(s-32.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.709171\) |
| Root analytic conductor: | \(0.891764\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.17719084i, 0.433826574i, -32.6110174i:\ ),\ 1)\) |
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
−23.21878, −20.84988, −19.98789, −19.16062, −17.93723, −14.77175, −13.75007, −11.77343, −10.06573, −9.28433, −7.41444, −3.45690, −2.21775, 3.19022, 5.64165, 7.97398, 8.37826, 9.49272, 12.91630, 14.42792, 15.80147, 16.57826, 18.48359, 19.97011, 21.75223, 24.50936