Dirichlet series
| L(s) = 1 | + (1.06 + 0.534i)2-s + (−0.177 − 0.405i)3-s + (−0.210 + 1.67i)4-s + (−0.517 − 1.38i)5-s + (0.0264 − 0.529i)6-s + (−0.390 + 0.483i)7-s + (−1.55 + 1.68i)8-s + (0.0449 − 0.261i)9-s + (0.186 − 1.76i)10-s + (−0.833 − 0.699i)11-s + (0.718 − 0.212i)12-s + (0.164 − 0.666i)13-s + (−0.676 + 0.308i)14-s + (−0.470 + 0.456i)15-s + (−2.16 − 0.402i)16-s + (−0.131 − 0.683i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+32.1i) \, \Gamma_{\R}(s+2.81i) \, \Gamma_{\R}(s-34.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(12.6993\) |
| Root analytic conductor: | \(2.33306\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (32.1693108i, 2.81555168i, -34.9848626i:\ ),\ 1)\) |
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
−23.57824, −22.81729, −21.68751, −19.53896, −18.43368, −15.47908, −14.49938, −13.08809, −10.94358, −10.03490, −6.55898, −4.43034, 0.19049, 3.17399, 4.63139, 5.74629, 7.91212, 8.76566, 12.02616, 12.68189, 13.64095, 15.66499, 16.72220, 18.17449, 20.48237, 21.68587, 23.18985, 24.06458