Dirichlet series
| L(s) = 1 | + (−0.250 + 0.433i)2-s + (0.150 − 0.786i)3-s + (−0.125 − 0.216i)4-s + (1.15 + 0.148i)5-s + (0.303 + 0.261i)6-s + (−0.186 − 0.826i)7-s +(0.125)·8-s + (−0.746 − 1.02i)9-s + (−0.354 + 0.464i)10-s + (0.166 + 0.581i)11-s + (−0.189 + 0.0658i)12-s + (−0.265 − 0.00552i)13-s + (0.404 + 0.126i)14-s + (0.291 − 0.889i)15-s + (−0.0312 + 0.0541i)16-s + (−0.711 + 1.25i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s-9.63i) \, \Gamma_{\R}(s-1.37i) \, \Gamma_{\R}(s+11.0i) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-0.5 - 0.866i$ |
| Analytic conductor: | \(2.19541\) |
| Root analytic conductor: | \(1.29968\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 4,\ (-9.63244453i, -1.3740602838i, 11.006504814i:\ ),\ -0.5 - 0.866i)\) |
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
−24.786086, −22.426191, −21.753908, −20.684415, −19.159041, −17.551677, −16.054131, −13.859784, −9.173289, −5.347232, −2.532102, 6.562632, 13.175860, 14.699394, 17.197402, 17.825519, 19.503737, 20.850137, 22.556509, 23.759265, 24.827213