Dirichlet series
| L(s) = 1 | + (0.914 − 0.0489i)2-s + (−1.26 − 2.14i)3-s + (−0.0808 − 0.138i)4-s + (0.709 + 0.335i)5-s + (−1.26 − 1.89i)6-s + (0.426 + 0.213i)7-s + (0.0811 − 0.122i)8-s + (−1.72 + 3.27i)9-s + (0.664 + 0.271i)10-s + (0.112 + 0.0954i)11-s + (−0.194 + 0.348i)12-s + (−0.322 − 0.735i)13-s + (0.400 + 0.174i)14-s + (−0.178 − 1.94i)15-s + (1.04 − 0.0345i)16-s + (−0.633 + 0.458i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-33.6i) \, \Gamma_{\R}(s-0.327i) \, \Gamma_{\R}(s+34.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.456816\) |
| Root analytic conductor: | \(0.770159\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-33.69003702i, -0.3278828322i, 34.01791986i:\ ),\ 1)\) |
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
−23.656298, −22.284158, −21.769640, −20.739395, −17.371140, −16.627711, −15.241495, −13.801224, −11.761812, −10.443039, −9.344898, −5.833414, −4.772076, −4.118442, 1.821187, 5.285525, 6.269639, 7.677919, 10.877755, 12.430288, 13.103893, 14.252088, 17.264258, 17.976543, 19.244914, 22.018059, 22.962160, 24.007357