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
−23.972, −21.628, −20.212, −19.403, −17.312, −15.969, −13.880, −11.854, −10.483, −9.311, −7.291, −4.611, −2.065, 2.623, 5.871, 7.327, 8.509, 10.556, 12.378, 14.532, 15.801, 17.397, 18.545, 20.278, 22.350, 24.460