Dirichlet series
| L(s) = 1 | + (0.617 + 0.0495i)2-s + (0.141 + 0.589i)3-s + (−0.238 + 0.110i)4-s + (−0.298 − 0.315i)5-s + (0.0577 + 0.370i)6-s + (−0.818 − 1.58i)7-s + (0.464 + 0.0564i)8-s + (−0.469 + 0.756i)9-s + (−0.168 − 0.209i)10-s + (0.328 − 0.0238i)11-s + (−0.0989 − 0.125i)12-s + (−0.566 − 0.469i)13-s + (−0.426 − 1.02i)14-s + (0.144 − 0.220i)15-s + (1.04 + 0.0272i)16-s + (−0.222 + 1.07i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+25.3i) \, \Gamma_{\R}(s+14.6i) \, \Gamma_{\R}(s-39.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(59.4917\) |
| Root analytic conductor: | \(3.90378\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (25.309589912i, 14.6099936176i, -39.91958353i:\ ),\ 1)\) |
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
−22.475631, −18.965903, −12.231714, −9.073121, −6.855292, −5.295719, −3.438097, −2.185344, −0.338337, 1.128208, 3.288580, 4.176550, 5.088250, 6.965400, 8.240604, 9.972101, 10.824912, 12.841058, 13.606892, 14.837365, 16.530716, 17.005929, 19.548440, 20.063503, 21.875933, 22.745470, 23.759133