Dirichlet series
| L(s) = 1 | + (−0.769 + 0.785i)2-s + (−0.406 − 0.149i)3-s + (0.743 − 0.423i)4-s + (−0.367 − 0.112i)5-s + (0.429 − 0.204i)6-s + (−0.695 + 1.08i)7-s + (−0.448 + 0.909i)8-s + (0.548 − 0.0279i)9-s + (0.371 − 0.201i)10-s + (0.575 − 0.180i)11-s + (−0.365 + 0.0609i)12-s + (−0.717 − 1.04i)13-s + (−0.319 − 1.38i)14-s + (0.132 + 0.100i)15-s + (−0.234 − 0.00761i)16-s + (0.711 − 0.328i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-32.4i) \, \Gamma_{\R}(s-9.36i) \, \Gamma_{\R}(s+41.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(51.2036\) |
| Root analytic conductor: | \(3.71335\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-32.4418788i, -9.36875716i, 41.8106358i:\ ),\ 1)\) |
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
−24.62830, −23.18886, −21.80623, −20.28658, −19.59154, −18.49681, −16.70322, −16.17720, −14.26814, −12.42485, −11.40623, −10.27017, −9.57967, −7.40479, −6.65622, −4.44459, −3.09793, −1.47571, −0.56683, 0.99649, 2.88372, 5.65279, 7.30889, 12.10678, 15.40713, 16.47399, 18.14805, 19.87028, 22.24987, 24.41085