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
−23.92655, −21.31474, −18.71735, −17.22242, −15.65833, −13.98548, −12.39532, −8.68267, −8.27470, −6.37466, −3.69605, 2.53740, 4.81404, 8.48506, 10.08741, 11.16191, 13.65356, 14.97695, 18.09025, 19.68844, 20.58567, 22.94262