Dirichlet series
| L(s) = 1 | + (−1.02 + 1.98i)2-s + (−0.871 − 0.754i)3-s + (−1.85 − 2.09i)4-s + (0.926 − 0.185i)5-s + (2.39 − 0.953i)6-s + (−0.0112 − 0.280i)7-s + (2.05 − 1.52i)8-s + (1.06 + 0.560i)9-s + (−0.583 + 2.02i)10-s + (0.538 + 0.492i)11-s + (0.0374 + 3.21i)12-s + (−0.0819 + 0.322i)13-s + (0.567 + 0.265i)14-s + (−0.947 − 0.537i)15-s + (2.13 + 1.81i)16-s + (0.0283 + 0.00926i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+28.5i) \, \Gamma_{\R}(s+12.0i) \, \Gamma_{\R}(s-40.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(56.2967\) |
| Root analytic conductor: | \(3.83260\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (28.52586324i, 12.0647034i, -40.59056664i:\ ),\ 1)\) |
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
−22.01359, −20.99790, −18.81703, −17.35424, −10.52583, −9.35135, −5.41918, −3.50500, −1.91712, −0.96595, 0.45418, 1.42637, 4.66883, 6.12711, 6.69815, 7.58926, 9.11078, 10.07772, 12.41694, 13.67078, 15.25445, 16.50832, 17.36992, 17.90735, 18.96675, 21.64434, 23.22054, 24.37312, 24.93710