Dirichlet series
| L(s) = 1 | + (1.15 − 0.437i)2-s + (−0.242 + 0.0944i)3-s + (−0.00902 − 1.44i)4-s + (−0.621 + 1.05i)5-s + (−0.239 + 0.215i)6-s + (−0.312 + 0.693i)7-s + (−1.17 − 1.67i)8-s + (0.292 + 0.0485i)9-s + (−0.257 + 1.49i)10-s + (0.738 − 0.148i)11-s + (0.139 + 0.351i)12-s + (−0.906 − 0.771i)13-s + (−0.0588 + 0.939i)14-s + (0.0512 − 0.314i)15-s + (−1.55 − 0.179i)16-s + (−0.752 − 0.185i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-29.7i) \, \Gamma_{\R}(s-0.113i) \, \Gamma_{\R}(s+29.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.127300\) |
| Root analytic conductor: | \(0.503049\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-29.7450486i, -0.113364765i, 29.8584134i:\ ),\ 1)\) |
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
−24.35631, −23.14445, −21.76818, −20.06128, −17.29589, −16.27795, −14.24453, −12.84017, −11.95404, −8.86200, −7.15532, −4.86186, −3.91605, 3.23517, 4.96832, 6.61394, 9.69279, 11.27055, 12.82811, 14.52441, 15.39008, 18.41396, 19.56989, 22.23172, 22.47116, 24.29327