Dirichlet series
| L(s) = 1 | + (0.0691 − 0.431i)2-s + (−0.745 − 1.43i)3-s + (−0.250 − 0.491i)4-s + (0.492 − 0.204i)5-s + (−0.669 + 0.222i)6-s + (−0.327 + 1.11i)7-s + (0.578 + 0.0743i)8-s + (−0.747 + 0.702i)9-s + (−0.0541 − 0.227i)10-s + (0.343 − 0.609i)11-s + (−0.516 + 0.725i)12-s + (−0.792 + 1.43i)13-s + (0.458 + 0.218i)14-s + (−0.659 − 0.553i)15-s + (−0.0536 − 0.534i)16-s + (0.356 − 0.353i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-29.0i) \, \Gamma_{\R}(s-0.381i) \, \Gamma_{\R}(s+29.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.448900\) |
| Root analytic conductor: | \(0.765684\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-29.090787498i, -0.38103922084i, 29.47182672i:\ ),\ 1)\) |
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
−22.774178, −22.062374, −20.222208, −17.196248, −16.734280, −15.062425, −13.167096, −10.708602, −9.909470, −7.401423, −5.252109, −3.944182, 1.819457, 5.482348, 6.885893, 9.306439, 11.532009, 12.626497, 14.081181, 16.791706, 18.447192, 19.277138, 21.847068, 23.526460, 24.947543