Dirichlet series
| L(s) = 1 | + (−1.15 + 1.90i)2-s + (0.336 − 0.107i)3-s + (−1.16 − 2.49i)4-s + (0.282 + 0.177i)5-s + (−0.183 + 0.767i)6-s + (−0.464 − 0.451i)7-s + (2.13 + 0.669i)8-s + (−0.234 − 0.179i)9-s + (−0.666 + 0.334i)10-s + (0.0796 − 0.0299i)11-s + (−0.659 − 0.716i)12-s + (−0.475 + 0.642i)13-s + (1.39 − 0.366i)14-s + (0.114 + 0.0295i)15-s + (−1.46 + 0.109i)16-s + (0.791 − 0.00586i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+18.1i) \, \Gamma_{\R}(s+0.492i) \, \Gamma_{\R}(s-18.6i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.298137\) |
| Root analytic conductor: | \(0.668044\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (18.1948153045508i, 0.49254810358012i, -18.6873634081308i:\ ),\ 1)\) |
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
−21.69423569291, −19.90628977327, −12.38648116997, −10.37195480581, −8.88186254645, −2.74848654512, 6.22826905980, 7.75481954982, 9.52111081005, 14.53233068545, 16.65949857790, 23.65383359484