Dirichlet series
| L(s) = 1 | + (−0.240 + 1.29i)2-s + (0.736 − 0.418i)3-s + (−1.36 + 0.669i)4-s + (−0.243 − 0.274i)5-s + (0.362 + 1.05i)6-s + (0.375 + 0.379i)7-s + (−1.25 − 1.92i)8-s + (−0.369 − 1.03i)9-s + (0.412 − 0.248i)10-s + (−0.353 − 0.843i)11-s + (−0.725 + 1.06i)12-s + (0.671 + 0.701i)13-s + (−0.580 + 0.393i)14-s + (−0.294 − 0.100i)15-s + (1.35 − 1.46i)16-s + (−0.218 + 1.08i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+26.6i) \, \Gamma_{\R}(s+0.329i) \, \Gamma_{\R}(s-26.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.289643\) |
| Root analytic conductor: | \(0.661638\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (26.65401432i, 0.3298547894i, -26.9838691i:\ ),\ 1)\) |
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
−22.94262, −20.58567, −19.68844, −18.09025, −14.97695, −13.65356, −11.16191, −10.08741, −8.48506, −4.81404, −2.53740, 3.69605, 6.37466, 8.27470, 8.68267, 12.39532, 13.98548, 15.65833, 17.22242, 18.71735, 21.31474, 23.92655