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
−24.278949288, −22.890264604, −21.920334804, −20.234411486, −17.414853804, −16.189015970, 6.028325425, 10.280168565, 11.648210792, 13.912595350, 19.295091037, 20.966010253, 22.422146922, 23.269117762