Dirichlet series
| L(s) = 1 | + (−0.663 − 0.786i)2-s + (0.133 + 0.371i)3-s + (0.484 + 0.257i)4-s + (0.111 − 0.562i)5-s + (0.204 − 0.351i)6-s + (0.143 − 0.687i)7-s + (−0.178 − 0.551i)8-s + (−0.253 + 0.470i)9-s + (−0.516 + 0.285i)10-s + (−0.294 + 0.477i)11-s + (−0.0311 + 0.214i)12-s + (−0.983 + 1.38i)13-s + (−0.636 + 0.343i)14-s + (0.223 − 0.0335i)15-s + (−0.455 − 0.490i)16-s + (−0.271 + 0.271i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-31.4i) \, \Gamma_{\R}(s-0.218i) \, \Gamma_{\R}(s+31.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.224242\) |
| Root analytic conductor: | \(0.607536\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-31.45942i, -0.2183829i, 31.677802i:\ ),\ 1)\) |
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
−24.460, −22.350, −20.278, −18.545, −17.397, −15.801, −14.532, −12.378, −10.556, −8.509, −7.327, −5.871, −2.623, 2.065, 4.611, 7.291, 9.311, 10.483, 11.854, 13.880, 15.969, 17.312, 19.403, 20.212, 21.628, 23.972