Dirichlet series
L(s) = 1 | + (−0.681 + 0.352i)2-s + (0.284 + 0.547i)3-s + (−0.687 − 0.480i)4-s + (−0.378 − 0.715i)5-s + (−0.386 − 0.273i)6-s + (−0.518 + 0.727i)7-s + (0.656 − 0.628i)8-s + (−0.292 + 0.311i)9-s + (0.509 + 0.354i)10-s + (−0.217 + 0.287i)11-s + (0.0674 − 0.512i)12-s + (−0.521 − 0.259i)13-s + (0.0975 − 0.678i)14-s + (0.284 − 0.410i)15-s + (0.0690 + 1.15i)16-s + (0.361 − 0.466i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.0i) \, \Gamma_{\R}(s+3.20i) \, \Gamma_{\R}(s-4.72i) \, \Gamma_{\R}(s-20.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
Degree: | \(4\) |
Conductor: | \(1\) |
Sign: | $1$ |
Analytic conductor: | \(4.37010\) |
Root analytic conductor: | \(1.44584\) |
Rational: | no |
Arithmetic: | no |
Primitive: | yes |
Self-dual: | no |
Selberg data: | \((4,\ 1,\ (22.057942217i, 3.20249067422i, -4.72506689836i, -20.5353659928i:\ ),\ 1)\) |
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
−23.4978571, −19.6325912, −18.4405063, −17.0301327, −14.4818902, −12.9873097, −10.9625074, −9.1009901, −7.4015726, −0.0067571, 8.5881260, 9.8174750, 12.7628598, 15.0632884, 16.6115206, 18.7199238, 22.6241674, 24.6391056