Dirichlet series
| L(s) = 1 | − 1.56·2-s − 0.00434·3-s + 0.635·4-s + 1.97·5-s + 0.00678·6-s − 0.760·7-s + 0.256·8-s − 0.717·9-s − 3.08·10-s + 0.218·11-s − 0.00276·12-s − 0.255·13-s + 1.18·14-s − 0.00858·15-s − 0.108·16-s − 0.676·17-s + 1.11·18-s + 1.01·19-s + 1.25·20-s + 0.00330·21-s − 0.341·22-s − 0.492·23-s − 0.00111·24-s + 1.56·25-s + 0.398·26-s + 0.00188·27-s − 0.482·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.9i) \, \Gamma_{\R}(s+9.18i) \, \Gamma_{\R}(s-19.9i) \, \Gamma_{\R}(s-9.18i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(21.5708\) |
| Root analytic conductor: | \(2.15509\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (19.98213190102i, 9.18145606512i, -19.98213190102i, -9.18145606512i:\ ),\ 1)\) |
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
−24.977383690, −22.187733048, −18.038235090, −17.032195673, −13.683152715, −9.655255845, −5.933277329, −2.380658862, −0.760354899, 0.760354899, 2.380658862, 5.933277329, 9.655255845, 13.683152715, 17.032195673, 18.038235090, 22.187733048, 24.977383690