Dirichlet series
| L(s) = 1 | − 1.29·2-s − 1.51·3-s + 0.333·4-s − 0.0818·5-s + 1.96·6-s − 0.784·7-s + 0.00796·8-s + 0.631·9-s + 0.105·10-s + 0.923·11-s − 0.505·12-s − 1.21·13-s + 1.01·14-s + 0.124·15-s + 0.215·16-s − 0.0602·17-s − 0.816·18-s + 0.414·19-s − 0.0272·20-s + 1.19·21-s − 1.19·22-s + 0.214·23-s − 0.0120·24-s − 1.20·25-s + 1.56·26-s + 0.0610·27-s − 0.261·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+16.7i) \, \Gamma_{\R}(s+6.16i) \, \Gamma_{\R}(s-16.7i) \, \Gamma_{\R}(s-6.16i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.85400\) |
| Root analytic conductor: | \(1.61802\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (16.78915308896i, 6.16376335886i, -16.78915308896i, -6.16376335886i:\ ),\ 1)\) |
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
−24.746566854, −23.070304673, −22.128152877, −19.358929434, −17.442493140, −11.831385964, −9.665235776, −0.455583474, 0.455583474, 9.665235776, 11.831385964, 17.442493140, 19.358929434, 22.128152877, 23.070304673, 24.746566854