Dirichlet series
| L(s) = 1 | + (0.692 − 0.204i)2-s + (−0.294 − 0.846i)3-s + (−0.255 − 0.488i)4-s + (−0.392 + 1.43i)5-s + (−0.377 − 0.525i)6-s + (−0.421 + 0.534i)7-s + (0.202 − 0.285i)8-s + (−0.335 − 0.347i)9-s + (0.0220 + 1.07i)10-s + (−0.819 − 0.398i)11-s + (−0.338 + 0.359i)12-s + (−0.505 − 0.728i)13-s + (−0.182 + 0.456i)14-s + (1.32 − 0.0904i)15-s + (0.850 − 0.0538i)16-s + (0.249 − 0.0831i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-26.2i) \, \Gamma_{\R}(s-0.126i) \, \Gamma_{\R}(s+26.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.103728\) |
| Root analytic conductor: | \(0.469856\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-26.241299732i, -0.126312068996i, 26.3676118i:\ ),\ 1)\) |
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
−23.49033, −21.77926, −20.06660, −16.84598, −16.03249, −13.70080, −12.31848, −10.02638, −8.03384, −5.01745, −4.25462, 3.11729, 5.76186, 7.42141, 10.37943, 12.15413, 13.69872, 15.20917, 18.14242, 19.20609, 22.23356, 23.37201