Dirichlet series
| L(s) = 1 | + (−1.01 − 1.46i)2-s + (−0.833 − 1.14i)3-s + (−0.0834 + 1.51i)4-s + (0.570 − 0.159i)5-s + (−0.820 + 2.37i)6-s + (0.127 + 0.107i)7-s + (0.124 − 1.41i)8-s + (0.225 + 0.761i)9-s + (−0.814 − 0.671i)10-s + (0.760 + 0.311i)11-s + (1.79 − 1.16i)12-s + (−0.609 + 0.521i)13-s + (0.0274 − 0.294i)14-s + (−0.658 − 0.518i)15-s + (−1.08 + 1.45i)16-s + (−0.294 + 0.0766i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.2i) \, \Gamma_{\R}(s-0.640i) \, \Gamma_{\R}(s+21.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.700851\) |
| Root analytic conductor: | \(0.888263\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-21.2549156293235i, -0.640424415083941i, 21.89534004440744i:\ ),\ 1)\) |
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
−24.82122607564, −17.49298761074, −16.45485600259, −15.05513242812, −10.87198101180, −9.21749102321, −6.96033708168, −5.32432841313, 1.72939774616, 6.76307100439, 9.62118194577, 11.46011767088, 12.61050409020, 17.43063291272, 18.94994153657, 24.70732933472