Dirichlet series
| L(s) = 1 | + (1.04 − 0.286i)2-s + (2.23 − 0.105i)3-s + (−0.0355 − 0.883i)4-s + (0.156 + 0.705i)5-s + (2.30 − 0.750i)6-s + (−0.121 − 0.589i)7-s + (−0.462 − 0.912i)8-s + (2.76 − 0.578i)9-s + (0.364 + 0.692i)10-s + (0.385 + 0.164i)11-s + (−0.172 − 1.97i)12-s + (0.163 + 0.281i)13-s + (−0.295 − 0.581i)14-s + (0.423 + 1.56i)15-s + (0.0849 − 0.173i)16-s + (1.04 − 0.0920i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-31.5i) \, \Gamma_{\R}(s-7.40i) \, \Gamma_{\R}(s+38.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(36.6512\) |
| Root analytic conductor: | \(3.32171\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-31.538134i, -7.4078318i, 38.945964i:\ ),\ 1)\) |
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
−24.0662, −22.1417, −21.0580, −20.1576, −18.8302, −16.6136, −15.0468, −14.2202, −13.3628, −12.4634, −9.5366, −8.5063, −7.8197, −5.3839, −3.9270, −3.1947, −1.8634, 1.4514, 3.0208, 3.9287, 9.7166, 13.2813, 14.2223, 14.9111, 18.8886, 19.9115, 21.3851, 23.2942