Dirichlet series
| L(s) = 1 | − 1.53·2-s + 1.83·3-s + 0.335·4-s + 0.883·5-s − 2.81·6-s − 0.0172·7-s + 1.03·8-s + 1.23·9-s − 1.35·10-s + 0.187·11-s + 0.616·12-s − 0.564·13-s + 0.0264·14-s + 1.62·15-s − 0.910·16-s + 0.280·17-s − 1.89·18-s − 0.0607·19-s + 0.296·20-s − 0.0317·21-s − 0.287·22-s − 0.0216·23-s + 1.89·24-s − 0.0454·25-s + 0.864·26-s + 0.163·27-s − 0.00579·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+21.5i) \, \Gamma_{\R}(s+2.26i) \, \Gamma_{\R}(s-21.5i) \, \Gamma_{\R}(s-2.26i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.49656\) |
| Root analytic conductor: | \(1.10604\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (21.569313052i, 2.26406194798i, -21.569313052i, -2.26406194798i:\ ),\ 1)\) |
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
−19.8788017, −18.5827755, −17.1535813, −14.6025983, −13.4927512, −9.8566951, −9.0845287, −8.0012161, 8.0012161, 9.0845287, 9.8566951, 13.4927512, 14.6025983, 17.1535813, 18.5827755, 19.8788017