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
−24.4467, −22.1438, −21.2170, −17.8587, −16.5891, −15.8288, −13.6052, −11.4967, −9.4777, −7.8621, −5.4939, −4.2981, 1.7257, 5.2609, 7.2321, 10.0450, 11.3284, 12.9001, 14.5223, 17.8309, 18.4176, 20.2225, 22.6292, 24.0082