Dirichlet series
| L(s) = 1 | + (−0.273 − 0.929i)2-s + (2.41 + 0.0124i)3-s + (−0.514 − 0.420i)4-s + (0.698 + 0.241i)5-s + (−0.650 − 2.25i)6-s + (0.0444 + 0.0721i)7-s + (−0.188 + 0.593i)8-s + (3.43 + 0.0724i)9-s + (0.0335 − 0.715i)10-s + (0.753 − 0.00325i)11-s + (−1.24 − 1.02i)12-s + (0.142 + 0.254i)13-s + (0.0549 − 0.0610i)14-s + (1.68 + 0.594i)15-s + (−0.202 − 0.553i)16-s + (0.355 + 0.295i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-27.0i) \, \Gamma_{\R}(s-8.44i) \, \Gamma_{\R}(s+35.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(32.6483\) |
| Root analytic conductor: | \(3.19610\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-27.03112124i, -8.449624804i, 35.48074604i:\ ),\ 1)\) |
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
−24.521127, −21.760696, −20.493467, −19.450091, −17.790278, −16.183411, −14.829413, −13.875929, −12.990884, −9.713471, −8.748065, −8.037534, −6.808884, −4.233718, −2.994496, −1.667741, 1.562219, 2.195010, 3.662287, 9.739470, 13.709518, 14.655617, 18.856917, 19.832776, 21.100459