Dirichlet series
| L(s) = 1 | + (−0.0542 + 0.915i)2-s + (−0.744 + 1.12i)3-s + (−0.780 + 0.816i)4-s + (0.317 − 0.550i)5-s + (−0.990 − 0.742i)6-s + (0.780 + 0.252i)7-s + (−0.545 − 0.759i)8-s + (0.0288 − 0.550i)9-s + (0.487 + 0.320i)10-s + (−0.393 − 0.945i)11-s + (−0.338 − 1.48i)12-s + (−0.282 + 0.536i)13-s + (−0.273 + 0.700i)14-s + (0.384 + 0.767i)15-s + (−0.119 − 0.213i)16-s + (−0.312 − 0.408i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+27.7i) \, \Gamma_{\R}(s+0.494i) \, \Gamma_{\R}(s-28.2i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.692098\) |
| Root analytic conductor: | \(0.884550\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (27.7352008i, 0.494437226i, -28.229638i:\ ),\ 1)\) |
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
−24.0082, −22.6292, −20.2225, −18.4176, −17.8309, −14.5223, −12.9001, −11.3284, −10.0450, −7.2321, −5.2609, −1.7257, 4.2981, 5.4939, 7.8621, 9.4777, 11.4967, 13.6052, 15.8288, 16.5891, 17.8587, 21.2170, 22.1438, 24.4467