Dirichlet series
| L(s) = 1 | + (1.21 + 0.429i)2-s + (−1.04 + 0.259i)3-s + (0.717 + 1.04i)4-s + (1.08 + 0.101i)5-s + (−1.38 − 0.134i)6-s + (−0.864 − 0.407i)7-s + (0.936 + 0.902i)8-s + (0.254 − 0.544i)9-s + (1.27 + 0.588i)10-s + (−0.189 + 0.111i)11-s + (−1.02 − 0.910i)12-s + (0.423 − 0.00317i)13-s + (−0.876 − 0.867i)14-s + (−1.16 + 0.174i)15-s + (1.00 + 0.896i)16-s + (−0.489 + 0.244i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+23.6i) \, \Gamma_{\R}(s+1.85i) \, \Gamma_{\R}(s-4.45i) \, \Gamma_{\R}(s-21.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.58174\) |
| Root analytic conductor: | \(1.26758\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (23.6793710964i, 1.853109050162i, -4.45930798328i, -21.0731721632i:\ ),\ 1)\) |
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
−22.77459779, −21.56786160, −19.44817917, −17.55502443, −15.96530633, −13.81073854, −13.04423133, −11.42069282, −10.01071908, −6.29371486, −5.21454419, 6.34638807, 10.46964051, 12.64873387, 13.86222725, 16.15875486, 17.32153507, 22.26510432, 23.25000004