Dirichlet series
| L(s) = 1 | + (−0.0391 − 0.0382i)2-s + (−0.0739 + 0.614i)3-s + (0.0392 − 0.0352i)4-s + (1.17 + 0.193i)5-s + (0.0263 − 0.0212i)6-s + (1.17 + 0.323i)7-s + (0.994 − 0.000119i)8-s + (−0.298 + 0.523i)9-s + (−0.0387 − 0.0526i)10-s + (−0.722 + 0.423i)11-s + (0.0187 + 0.0267i)12-s + (−0.408 + 1.32i)13-s + (−0.0336 − 0.0576i)14-s + (−0.206 + 0.711i)15-s + (−0.0778 − 0.0790i)16-s + (0.381 + 0.316i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-25.7i) \, \Gamma_{\R}(s-17.1i) \, \Gamma_{\R}(s+42.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(76.6510\) |
| Root analytic conductor: | \(4.24788\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-25.75536556i, -17.192718445i, 42.948084006i:\ ),\ 1)\) |
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
−24.869592, −23.820271, −22.413852, −20.983517, −20.094836, −18.198757, −17.683167, −16.459232, −14.549191, −13.637141, −12.601554, −10.976500, −9.954163, −8.237273, −7.349005, −5.733745, −4.879062, −2.846324, −1.526392, −0.812123, 1.543827, 2.291767, 4.549010, 5.430702, 7.568371, 9.583998, 11.000684, 13.971250, 21.496079