Dirichlet series
| L(s) = 1 | + (−0.238 − 0.946i)2-s + (0.409 + 0.360i)3-s + (−0.600 − 0.495i)4-s + (0.00893 − 0.403i)5-s + (0.244 − 0.473i)6-s + (−0.991 + 1.25i)7-s + (−0.278 + 0.686i)8-s + (−0.372 + 0.656i)9-s + (−0.384 + 0.0877i)10-s + (0.0711 − 0.824i)11-s + (−0.0670 − 0.419i)12-s + (−0.598 + 0.0805i)13-s + (1.42 + 0.639i)14-s + (0.149 − 0.162i)15-s + (−0.133 − 0.395i)16-s + (0.671 − 0.292i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-25.2i) \, \Gamma_{\R}(s-0.243i) \, \Gamma_{\R}(s+25.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.164569\) |
| Root analytic conductor: | \(0.548003\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-25.27961566i, -0.2430168932i, 25.52263256i:\ ),\ 1)\) |
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
−22.812865, −19.882193, −17.687387, −16.327596, −14.304332, −12.718105, −9.820639, −7.744307, −6.757323, −3.647261, 2.721292, 5.404222, 8.838084, 10.034902, 11.938378, 13.965832, 16.042992, 18.823934, 19.919083, 22.010794