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
−25.00930739, −24.07516619, −21.31734938, −18.01935370, −16.02907799, −14.80357555, −11.82049665, −10.72054792, −8.41897476, −6.15059104, 1.70838291, 11.51000602, 17.90505562, 20.18471905, 21.21464510, 23.73325788, 24.47016955