Dirichlet series
| L(s) = 1 | + (1.28 − 0.0423i)2-s + (−0.288 − 0.280i)3-s + (0.359 − 0.150i)4-s + (−0.0432 − 0.854i)5-s + (−0.381 − 0.347i)6-s + (−0.287 + 0.403i)7-s + (−0.190 − 0.208i)8-s + (0.292 − 0.118i)9-s + (−0.0917 − 1.09i)10-s + (−0.246 − 0.391i)11-s + (−0.146 − 0.0575i)12-s + (−1.13 − 1.83i)13-s + (−0.350 + 0.530i)14-s + (−0.227 + 0.258i)15-s + (0.561 − 0.123i)16-s + (0.840 + 0.0416i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+29.4i) \, \Gamma_{\R}(s+3.03i) \, \Gamma_{\R}(s-32.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(11.6429\) |
| Root analytic conductor: | \(2.26649\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (29.41987002i, 3.039198812i, -32.45906884i:\ ),\ 1)\) |
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
−23.714169, −22.487941, −21.452914, −19.024625, −16.812558, −14.838541, −13.722706, −12.019955, −10.090418, −6.950056, −4.685630, 0.764420, 3.185940, 4.910034, 5.748282, 7.981056, 10.031348, 12.418132, 12.736384, 14.348671, 15.886867, 17.626933, 19.556330, 21.285755, 22.542985, 23.721749, 24.997488