Dirichlet series
| L(s) = 1 | + (−0.386 − 0.573i)2-s + (0.302 + 0.471i)3-s + (0.206 − 0.130i)4-s + (0.418 − 0.321i)5-s + (0.153 − 0.355i)6-s + (1.08 − 0.204i)7-s + (0.367 − 0.0679i)8-s + (−0.433 + 0.756i)9-s + (−0.346 − 0.115i)10-s + (0.278 − 0.794i)11-s + (0.123 + 0.0578i)12-s + (−0.612 + 0.511i)13-s + (−0.537 − 0.544i)14-s + (0.278 + 0.0999i)15-s + (−0.562 − 0.926i)16-s + (0.0943 + 0.442i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-24.9i) \, \Gamma_{\R}(s-8.60i) \, \Gamma_{\R}(s+33.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(28.9911\) |
| Root analytic conductor: | \(3.07200\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-24.9344938i, -8.60479548i, 33.5392892i:\ ),\ 1)\) |
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
−24.78669, −23.00501, −21.13236, −19.69330, −17.85241, −17.25955, −15.25173, −14.17675, −12.45704, −10.85112, −8.99240, −7.68502, −6.70887, −4.84862, −2.64421, −1.29723, 1.01513, 2.30993, 4.75764, 10.93024, 14.19331, 16.85399, 19.45506, 21.20392