Dirichlet series
| L(s) = 1 | + (−1.04 − 1.63i)2-s + (0.919 − 0.441i)3-s + (−0.553 + 1.77i)4-s + (0.0996 + 0.740i)5-s + (−1.67 − 1.04i)6-s + (0.859 − 0.448i)7-s + (0.715 − 0.940i)8-s + (−0.268 − 1.25i)9-s + (1.10 − 0.934i)10-s + (−0.102 − 0.689i)11-s + (0.273 + 1.87i)12-s + (−0.117 + 0.780i)13-s + (−1.62 − 0.938i)14-s + (0.418 + 0.636i)15-s + (−0.999 + 0.926i)16-s + (−0.755 + 0.399i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+25.6i) \, \Gamma_{\R}(s+8.89i) \, \Gamma_{\R}(s-34.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(31.8419\) |
| Root analytic conductor: | \(3.16956\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (25.69438766i, 8.89310018i, -34.58748784i:\ ),\ 1)\) |
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
−24.894383, −20.316810, −17.681054, −16.052150, −14.621057, −8.086901, −5.201758, −2.078416, 0.283652, 1.643432, 2.557360, 3.635363, 6.799801, 8.426380, 9.434454, 10.949108, 11.646024, 13.527488, 14.894409, 17.524206, 18.586952, 19.581298, 20.498204, 21.590367, 24.075284