Dirichlet series
| L(s) = 1 | + (0.806 + 0.468i)2-s + (−0.189 + 0.530i)3-s + (−0.375 + 1.22i)4-s + (−0.660 + 1.39i)5-s + (−0.400 + 0.339i)6-s + (0.0184 − 0.229i)7-s + (−0.744 + 0.810i)8-s + (−0.0563 + 0.329i)9-s + (−1.18 + 0.812i)10-s + (0.637 + 0.677i)11-s + (−0.577 − 0.430i)12-s + (−0.778 − 1.20i)13-s + (0.122 − 0.176i)14-s + (−0.612 − 0.613i)15-s + (−0.443 − 0.388i)16-s + (0.0324 − 0.0323i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.4i) \, \Gamma_{\R}(s+5.77i) \, \Gamma_{\R}(s-38.1i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(28.7762\) |
| Root analytic conductor: | \(3.06439\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.415328i, 5.7737672i, -38.189096i:\ ),\ 1)\) |
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
−24.2414, −23.7357, −21.9301, −20.0266, −18.8540, −16.5742, −14.4891, −12.9315, −11.8000, −9.1375, −4.3037, −1.3712, −0.3510, 2.9260, 3.9283, 5.0925, 6.9898, 7.9570, 10.0879, 11.4854, 12.9219, 14.4483, 15.1910, 16.7922, 18.0014, 19.8045, 21.7504, 22.5349, 23.1374