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
−24.997488, −23.721749, −22.542985, −21.285755, −19.556330, −17.626933, −15.886867, −14.348671, −12.736384, −12.418132, −10.031348, −7.981056, −5.748282, −4.910034, −3.185940, −0.764420, 4.685630, 6.950056, 10.090418, 12.019955, 13.722706, 14.838541, 16.812558, 19.024625, 21.452914, 22.487941, 23.714169