Dirichlet series
| L(s) = 1 | + (0.556 + 0.928i)2-s + (−0.579 + 0.0446i)3-s + (0.179 + 1.03i)4-s + (0.652 − 0.534i)5-s + (−0.363 − 0.512i)6-s + (−0.493 − 0.0183i)7-s + (0.104 + 0.491i)8-s + (−0.572 − 0.0517i)9-s + (0.859 + 0.308i)10-s + (0.130 + 0.335i)11-s + (−0.149 − 0.590i)12-s + (0.259 − 0.340i)13-s + (−0.257 − 0.468i)14-s + (−0.354 + 0.338i)15-s + (−0.0961 + 1.12i)16-s + (0.282 − 0.246i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.8i) \, \Gamma_{\R}(s+2.27i) \, \Gamma_{\R}(s-6.03i) \, \Gamma_{\R}(s-13.1i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.92978\) |
| Root analytic conductor: | \(1.17862\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (16.89972715592i, 2.27258771492i, -6.03583588968i, -13.13647898116i:\ ),\ 1)\) |
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
−23.269117762, −22.422146922, −20.966010253, −19.295091037, −13.912595350, −11.648210792, −10.280168565, −6.028325425, 16.189015970, 17.414853804, 20.234411486, 21.920334804, 22.890264604, 24.278949288