Dirichlet series
| L(s) = 1 | + (1.17 + 0.0364i)2-s + (0.431 + 0.505i)3-s + (0.202 + 0.122i)4-s + (0.516 + 0.193i)5-s + (0.487 + 0.609i)6-s + (0.00587 + 0.451i)7-s + (−0.145 + 0.150i)8-s + (−0.500 + 0.941i)9-s + (0.599 + 0.245i)10-s + (1.01 − 0.215i)11-s + (0.0255 + 0.155i)12-s + (−0.707 − 1.61i)13-s + (−0.00956 + 0.529i)14-s + (0.125 + 0.344i)15-s + (0.755 + 0.0721i)16-s + (−0.875 + 0.519i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+25.8i) \, \Gamma_{\R}(s+1.72i) \, \Gamma_{\R}(s-27.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.81697\) |
| Root analytic conductor: | \(1.68885\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (25.862536668i, 1.72423935728i, -27.586776026i:\ ),\ 1)\) |
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
−23.6614140, −21.7265144, −19.6601509, −17.2736788, −14.4709825, −13.6608526, −12.0633888, −9.2343795, −6.5969813, −4.3216340, 2.7213062, 4.4231594, 5.8452495, 8.4726623, 10.4177839, 12.6452563, 14.0291738, 15.0522595, 17.2242012, 19.7052444, 21.7323801, 22.5304690