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
−19.78711, −17.11155, −14.58112, −13.07666, −11.36647, −9.08364, −7.79507, −6.15820, −4.39230, −1.15095, 3.17888, 4.83663, 7.06229, 12.21164, 14.11404, 16.31873, 19.72178, 24.84700