Dirichlet series
| L(s) = 1 | + (−0.249 + 0.433i)2-s + (−0.322 − 0.951i)3-s + (−1.12 − 0.216i)4-s + (−0.125 − 0.216i)5-s + (0.492 + 0.0981i)6-s + (−0.929 − 0.803i)7-s + (0.624 − 0.866i)8-s + (−0.478 − 0.337i)9-s +(0.125)·10-s + (0.256 − 0.00189i)11-s + (0.157 + 1.14i)12-s + (1.31 + 0.423i)13-s + (0.580 − 0.201i)14-s + (−0.165 + 0.188i)15-s + (1.34 − 0.595i)16-s + (−0.475 + 0.0734i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+18.2i) \, \Gamma_{\R}(s+0.245i) \, \Gamma_{\R}(s-18.4i) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-0.5 - 0.866i$ |
| Analytic conductor: | \(0.348657\) |
| Root analytic conductor: | \(0.703827\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 4,\ (18.24250194i, 0.245580562i, -18.4880825i:\ ),\ -0.5 - 0.866i)\) |
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
−24.84700, −19.72178, −16.31873, −14.11404, −12.21164, −7.06229, −4.83663, −3.17888, 1.15095, 4.39230, 6.15820, 7.79507, 9.08364, 11.36647, 13.07666, 14.58112, 17.11155, 19.78711