Dirichlet series
| L(s) = 1 | + (0.677 − 0.839i)2-s + (−0.0660 − 0.190i)3-s + (−0.217 − 1.13i)4-s + (−0.722 + 0.445i)5-s + (−0.204 − 0.0732i)6-s + (0.161 − 0.260i)7-s + (−0.404 + 0.227i)8-s + (−0.335 + 0.0251i)9-s + (−0.115 + 0.908i)10-s + (0.155 − 2.12i)11-s + (−0.201 + 0.116i)12-s + (1.19 + 0.708i)13-s + (−0.108 − 0.311i)14-s + (0.132 + 0.107i)15-s + (0.0739 + 0.461i)16-s + (0.433 + 0.112i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.7i) \, \Gamma_{\R}(s-1.14i) \, \Gamma_{\R}(s+7.60i) \, \Gamma_{\R}(s+17.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.02260\) |
| Root analytic conductor: | \(1.19255\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-23.7804846144i, -1.141485214492i, 7.60977421348i, 17.31219561542i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.39664408, −22.56444137, −20.58138783, −15.40137552, −12.61591034, −4.29441687, 3.69633164, 5.94476917, 8.41565166, 10.97961379, 11.39737080, 13.52308045, 14.44034919, 16.38697267, 18.70071824, 19.66452990, 21.37405546, 23.35958806