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
−23.25000004, −22.26510432, −17.32153507, −16.15875486, −13.86222725, −12.64873387, −10.46964051, −6.34638807, 5.21454419, 6.29371486, 10.01071908, 11.42069282, 13.04423133, 13.81073854, 15.96530633, 17.55502443, 19.44817917, 21.56786160, 22.77459779