Dirichlet series
| L(s) = 1 | + (−0.654 − 1.40i)2-s + (−0.564 + 0.328i)3-s + (−0.880 + 0.434i)4-s + (−0.832 − 0.868i)5-s + (0.829 + 0.575i)6-s + (−0.665 + 0.239i)7-s + (−0.207 + 0.949i)8-s + (0.774 − 0.0420i)9-s + (−0.671 + 1.73i)10-s + (0.253 + 0.262i)11-s + (0.354 − 0.533i)12-s + (0.973 + 0.114i)13-s + (0.771 + 0.775i)14-s + (0.754 + 0.216i)15-s + (0.844 − 0.213i)16-s + (−0.653 − 0.714i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.1i) \, \Gamma_{\R}(s-4.28i) \, \Gamma_{\R}(s+27.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(10.9480\) |
| Root analytic conductor: | \(2.22047\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-23.166123532148i, -4.2813077502248i, 27.447431282372i:\ ),\ 1)\) |
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
−23.9771509051, −22.3795049884, −19.0261008756, −17.7636211353, −16.2775917339, −15.2820971771, −12.9378568518, −10.8769450805, −8.6370625433, −7.0461689821, −6.5135525540, −3.7826294016, −0.2696089954, 1.1669040301, 9.2285464564, 10.9464566127, 12.4602151149, 16.1187632499, 18.7113196733, 20.3766951996