Dirichlet series
| L(s) = 1 | − 1.31·2-s + 1.41·3-s + 0.00480·4-s + 0.0946·5-s − 1.86·6-s − 0.485·7-s + 0.956·8-s + 0.205·9-s − 0.124·10-s + 1.03·11-s + 0.00681·12-s + 0.789·13-s + 0.639·14-s + 0.134·15-s − 0.532·16-s + 0.480·17-s − 0.270·18-s + 1.34·19-s + 0.000400·20-s − 0.687·21-s − 1.36·22-s + 0.929·23-s + 1.35·24-s − 0.419·25-s − 1.04·26-s − 0.849·27-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.2i) \, \Gamma_{\R}(s+8.38i) \, \Gamma_{\R}(s-17.2i) \, \Gamma_{\R}(s-8.38i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(13.3876\) |
| Root analytic conductor: | \(1.91282\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.23810733754i, 8.3857414742i, -17.23810733754i, -8.3857414742i:\ ),\ 1)\) |
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
−24.94257939, −22.64737223, −20.34214760, −19.01890417, −14.01555238, −9.18304196, −3.25008280, −1.01144200, 1.01144200, 3.25008280, 9.18304196, 14.01555238, 19.01890417, 20.34214760, 22.64737223, 24.94257939