Dirichlet series
| L(s) = 1 | + (0.132 − 0.775i)2-s + (0.193 − 0.111i)3-s + (−0.716 − 0.980i)4-s + (0.531 − 0.0341i)5-s + (−0.0609 − 0.165i)6-s + (−0.475 + 0.0887i)7-s + (−0.474 + 0.425i)8-s + (−0.168 − 0.154i)9-s + (0.0439 − 0.416i)10-s + (0.377 − 0.101i)11-s + (−0.248 − 0.110i)12-s + (−0.952 + 1.87i)13-s + (0.00586 + 0.380i)14-s + (0.0992 − 0.0659i)15-s + (−0.266 + 0.334i)16-s + (0.0114 + 0.249i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-33.3i) \, \Gamma_{\R}(s-0.287i) \, \Gamma_{\R}(s+33.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.361854\) |
| Root analytic conductor: | \(0.712598\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-33.346206048i, -0.28779360564i, 33.633999654i:\ ),\ 1)\) |
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
−24.9687636, −22.7670560, −21.8147025, −19.9082960, −17.7756145, −16.8583806, −15.2933818, −13.6691644, −12.5527386, −10.0758829, −8.4474534, −7.0550153, −5.1591474, −3.2107012, 2.1030531, 4.5190985, 6.4105581, 9.1065412, 10.1419550, 11.8701419, 13.6042738, 14.5963986, 16.8951882, 18.7968553, 19.6213297, 21.3283447, 22.7534660, 24.3380005