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
−24.007357, −22.962160, −22.018059, −19.244914, −17.976543, −17.264258, −14.252088, −13.103893, −12.430288, −10.877755, −7.677919, −6.269639, −5.285525, −1.821187, 4.118442, 4.772076, 5.833414, 9.344898, 10.443039, 11.761812, 13.801224, 15.241495, 16.627711, 17.371140, 20.739395, 21.769640, 22.284158, 23.656298