Dirichlet series
| L(s) = 1 | + (−0.852 − 0.418i)2-s + (0.674 + 0.331i)3-s + (1.40 + 0.294i)4-s + (−0.309 − 0.474i)5-s + (−0.436 − 0.564i)6-s + (0.805 − 0.238i)7-s + (−0.974 − 0.837i)8-s + (−0.329 + 0.778i)9-s + (0.0651 + 0.533i)10-s + (−0.388 − 0.416i)11-s + (0.848 + 0.663i)12-s + (0.329 − 0.313i)13-s + (−0.786 − 0.133i)14-s + (−0.0511 − 0.422i)15-s + (0.947 + 0.367i)16-s + (−0.922 + 0.336i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-16.0i) \, \Gamma_{\R}(s-1.98i) \, \Gamma_{\R}(s+18.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.27539\) |
| Root analytic conductor: | \(1.31528\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-16.05436164407458i, -1.983654577143466i, 18.03801622121804i:\ ),\ 1)\) |
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
−24.084551066051, −20.399207031503, −15.053425806653, −11.268693372450, −8.429044761552, −6.887924121889, −2.322816219858, 8.198816052344, 10.914973844004, 19.730943822435, 20.936000401879, 24.379462767832