Dirichlet series
| L(s) = 1 | + (−0.486 + 1.43i)2-s + (−0.368 + 0.539i)3-s + (−1.34 + 0.0384i)4-s + (0.820 − 0.0404i)5-s + (−0.597 − 0.793i)6-s + (0.526 − 0.580i)7-s + (−0.707 − 1.95i)8-s + (0.213 + 0.141i)9-s + (−0.341 + 1.20i)10-s + (−0.222 + 0.542i)11-s + (0.476 − 0.742i)12-s + (−0.00781 − 0.372i)13-s + (0.579 + 1.04i)14-s + (−0.280 + 0.457i)15-s + (1.96 − 0.548i)16-s + (−0.193 + 0.905i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+23.8i) \, \Gamma_{\R}(s+0.532i) \, \Gamma_{\R}(s-24.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.598827\) |
| Root analytic conductor: | \(0.842882\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (23.852215973354i, 0.53206975702346i, -24.384285730376i:\ ),\ 1)\) |
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
−21.0506422343, −18.8533061940, −17.8451386177, −14.1055568007, −12.1037449797, −10.6005877392, −9.0350697691, −5.6276509244, −1.9798977494, 4.6245313558, 6.6442286091, 8.3324951671, 10.1225202594, 13.1973516289, 15.3207428713, 16.9361599090, 17.7441553744, 21.6525215441