Dirichlet series
| L(s) = 1 | + (−0.250 − 0.433i)2-s + (−0.788 + 1.44i)3-s + (−0.125 + 0.216i)4-s + (1.10 − 0.0994i)5-s + (0.824 − 0.0211i)6-s + (−0.971 − 0.969i)7-s +(0.125)·8-s + (−0.692 − 0.835i)9-s + (−0.320 − 0.455i)10-s + (1.12 − 0.373i)11-s + (−0.215 − 0.351i)12-s + (0.149 − 0.873i)13-s + (−0.176 + 0.662i)14-s + (−0.729 + 1.68i)15-s + (−0.0312 − 0.0541i)16-s + (0.462 + 0.0200i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-13.5i) \, \Gamma_{\R}(s-0.409i) \, \Gamma_{\R}(s+13.9i) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-0.5 + 0.866i$ |
| Analytic conductor: | \(0.454450\) |
| Root analytic conductor: | \(0.768827\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 4,\ (-13.54933871i, -0.4090616126i, 13.958400322i:\ ),\ -0.5 + 0.866i)\) |
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
−24.187310, −22.911382, −21.737101, −19.207409, −18.195760, −16.990900, −12.286437, −9.302532, −6.645868, −6.069471, −1.780465, 3.920999, 5.926865, 9.624630, 10.553687, 16.429479, 17.346896, 19.748777, 21.213934, 22.089463, 22.994143