Dirichlet series
| L(s) = 1 | + 1.49·2-s + 0.174·3-s + 0.225·4-s + 0.983·5-s + 0.260·6-s − 0.0443·7-s − 1.15·8-s − 1.12·9-s + 1.46·10-s + 1.29·11-s + 0.0393·12-s − 0.636·13-s − 0.0661·14-s + 0.172·15-s − 0.947·16-s + 0.600·17-s − 1.68·18-s + 0.901·19-s + 0.221·20-s − 0.00776·21-s + 1.92·22-s − 0.571·23-s − 0.201·24-s + 0.0492·25-s − 0.949·26-s − 0.225·27-s − 0.00998·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.1i) \, \Gamma_{\R}(s+3.52i) \, \Gamma_{\R}(s-20.1i) \, \Gamma_{\R}(s-3.52i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.20975\) |
| Root analytic conductor: | \(1.33849\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (20.1486283826i, 3.52286413626i, -20.1486283826i, -3.52286413626i:\ ),\ 1)\) |
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
−24.53113405, −22.83845310, −21.97028951, −17.33413669, −14.37120895, −13.88120811, −12.14703455, −9.31956767, −5.67900470, 5.67900470, 9.31956767, 12.14703455, 13.88120811, 14.37120895, 17.33413669, 21.97028951, 22.83845310, 24.53113405