Dirichlet series
| L(s) = 1 | + 0.0170·2-s + 0.893·3-s + 0.480·4-s − 0.439·5-s + 0.0152·6-s − 0.597·7-s + 0.0334·8-s + 0.713·9-s − 0.00747·10-s + 0.597·11-s + 0.429·12-s − 1.36·13-s − 0.0101·14-s − 0.392·15-s − 0.768·16-s + 1.04·17-s + 0.0121·18-s + 1.55·19-s − 0.211·20-s − 0.533·21-s + 0.0101·22-s − 1.01·23-s + 0.0298·24-s − 0.205·25-s − 0.0232·26-s + 1.45·27-s − 0.287·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.0i) \, \Gamma_{\R}(s+3.08i) \, \Gamma_{\R}(s-19.0i) \, \Gamma_{\R}(s-3.08i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.20087\) |
| Root analytic conductor: | \(1.21800\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (19.08484348882i, 3.08349605156i, -19.08484348882i, -3.08349605156i:\ ),\ 1)\) |
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
−24.265938649, −22.270037054, −20.124383080, −16.010264709, −14.269820267, −12.123704515, −9.631455702, −7.399219143, 7.399219143, 9.631455702, 12.123704515, 14.269820267, 16.010264709, 20.124383080, 22.270037054, 24.265938649