Dirichlet series
| L(s) = 1 | + (−0.404 + 0.0413i)2-s + (−0.177 + 0.424i)3-s + (1.29 − 0.0334i)4-s + (0.0685 + 0.218i)5-s + (0.0541 − 0.179i)6-s + (1.21 + 0.467i)7-s + (−1.38 + 0.0722i)8-s + (0.0994 − 0.150i)9-s + (−0.0367 − 0.0856i)10-s + (0.0579 − 0.689i)11-s + (−0.214 + 0.553i)12-s + (0.684 − 0.335i)13-s + (−0.509 − 0.139i)14-s + (−0.105 − 0.00974i)15-s + (1.17 − 0.124i)16-s + (−0.421 + 0.396i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-20.4i) \, \Gamma_{\R}(s-2.29i) \, \Gamma_{\R}(s+7.14i) \, \Gamma_{\R}(s+15.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.30542\) |
| Root analytic conductor: | \(1.34836\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-20.4354725578i, -2.29124298714i, 7.1433460158i, 15.5833695292i:\ ),\ 1)\) |
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
−24.47016955, −23.73325788, −21.21464510, −20.18471905, −17.90505562, −11.51000602, −1.70838291, 6.15059104, 8.41897476, 10.72054792, 11.82049665, 14.80357555, 16.02907799, 18.01935370, 21.31734938, 24.07516619, 25.00930739