Dirichlet series
| L(s) = 1 | + 0.273·2-s + 0.972·3-s + 0.945·4-s + 1.40·5-s + 0.266·6-s − 0.955·7-s + 0.770·8-s − 0.352·9-s + 0.383·10-s + 0.448·11-s + 0.919·12-s + 0.599·13-s − 0.261·14-s + 1.36·15-s + 0.108·16-s + 0.345·17-s − 0.0963·18-s + 0.0499·19-s + 1.32·20-s − 0.929·21-s + 0.122·22-s − 0.400·23-s + 0.749·24-s + 0.699·25-s + 0.163·26-s − 0.632·27-s − 0.903·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.5i) \, \Gamma_{\R}(s+9.57i) \, \Gamma_{\R}(s-16.5i) \, \Gamma_{\R}(s-9.57i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(16.0951\) |
| Root analytic conductor: | \(2.00296\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.54574513796i, 9.57822088538i, -16.54574513796i, -9.57822088538i:\ ),\ 1)\) |
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
−24.801004354, −22.608556513, −21.003536268, −19.761884281, −13.777028950, −6.233022843, −3.049803533, −1.812375219, 1.812375219, 3.049803533, 6.233022843, 13.777028950, 19.761884281, 21.003536268, 22.608556513, 24.801004354