Dirichlet series
| L(s) = 1 | + 0.0474·2-s + 0.0165·3-s + 0.324·4-s + 1.62·5-s + 0.000784·6-s − 0.0806·7-s + 0.0781·8-s + 1.07·9-s + 0.0769·10-s − 0.801·11-s + 0.00537·12-s + 0.557·13-s − 0.00382·14-s + 0.0268·15-s − 0.889·16-s + 1.08·17-s + 0.0508·18-s − 0.802·19-s + 0.526·20-s − 0.00133·21-s − 0.0380·22-s − 0.397·23-s + 0.00129·24-s + 0.365·25-s + 0.0264·26-s + 0.0519·27-s − 0.0261·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.1i) \, \Gamma_{\R}(s+6.05i) \, \Gamma_{\R}(s-16.1i) \, \Gamma_{\R}(s-6.05i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.16091\) |
| Root analytic conductor: | \(1.57547\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.1939597336i, 6.0588976646i, -16.1939597336i, -6.0588976646i:\ ),\ 1)\) |
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
−23.62527046, −21.71948245, −20.86917585, −18.32456959, −13.31444438, −10.07695978, −1.87972095, 1.87972095, 10.07695978, 13.31444438, 18.32456959, 20.86917585, 21.71948245, 23.62527046