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
−24.947543, −23.526460, −21.847068, −19.277138, −18.447192, −16.791706, −14.081181, −12.626497, −11.532009, −9.306439, −6.885893, −5.482348, −1.819457, 3.944182, 5.252109, 7.401423, 9.909470, 10.708602, 13.167096, 15.062425, 16.734280, 17.196248, 20.222208, 22.062374, 22.774178