Dirichlet series
| L(s) = 1 | + (0.195 + 0.815i)2-s + (−0.633 + 1.04i)3-s + (−0.822 + 1.13i)4-s + (−0.515 + 0.0315i)5-s + (−0.978 − 0.312i)6-s + (1.11 − 0.291i)7-s + (−0.789 − 0.449i)8-s + (−0.0629 − 0.280i)9-s + (−0.126 − 0.414i)10-s + (0.695 + 0.692i)11-s + (−0.667 − 1.58i)12-s + (0.194 − 0.386i)13-s + (0.455 + 0.852i)14-s + (0.293 − 0.560i)15-s + (−0.357 − 0.809i)16-s + (−0.737 − 1.15i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.3i) \, \Gamma_{\R}(s+0.524i) \, \Gamma_{\R}(s-32.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.06392\) |
| Root analytic conductor: | \(1.02086\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.3576548i, 0.524630384i, -32.8822852i:\ ),\ 1)\) |
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
−24.08206, −23.65237, −21.81644, −19.74212, −18.70825, −17.54572, −14.88163, −13.40097, −11.81673, −10.92546, −8.66131, −6.41951, −4.64031, −1.44392, 4.12738, 4.83633, 7.22105, 8.62947, 10.66423, 12.24573, 14.24639, 15.75845, 16.80078, 17.88864, 20.67325, 22.08569, 23.02861, 24.98828