Dirichlet series
| L(s) = 1 | + 2.85·2-s + 0.324·3-s + 4.18·4-s − 0.451·5-s + 0.924·6-s + 0.0784·7-s + 3.54·8-s + 0.214·9-s − 1.28·10-s + 0.000797·11-s + 1.35·12-s − 0.312·13-s + 0.223·14-s − 0.146·15-s + 0.756·16-s + 0.177·17-s + 0.611·18-s − 0.0726·19-s − 1.88·20-s + 0.0254·21-s + 0.00227·22-s + 0.852·23-s + 1.15·24-s − 0.699·25-s − 0.890·26-s + 0.429·27-s + 0.328·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+20.2i) \, \Gamma_{\R}(s+6.20i) \, \Gamma_{\R}(s-20.2i) \, \Gamma_{\R}(s-6.20i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(10.0948\) |
| Root analytic conductor: | \(1.78248\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (20.2316855252i, 6.20716677906i, -20.2316855252i, -6.20716677906i:\ ),\ 1)\) |
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
−23.86179517, −22.92254679, −21.52866298, −15.50486062, −14.37050950, −13.11510497, −11.83499912, −4.45784643, −2.94311851, 2.94311851, 4.45784643, 11.83499912, 13.11510497, 14.37050950, 15.50486062, 21.52866298, 22.92254679, 23.86179517