Dirichlet series
| L(s) = 1 | − 2.11·2-s − 0.424·3-s + 2.20·4-s + 0.924·5-s + 0.896·6-s − 0.508·7-s − 1.99·8-s − 0.831·9-s − 1.95·10-s + 0.335·11-s − 0.935·12-s − 0.374·13-s + 1.07·14-s − 0.392·15-s + 2.69·16-s − 2.60·17-s + 1.75·18-s − 0.167·19-s + 2.03·20-s + 0.215·21-s − 0.708·22-s − 1.08·23-s + 0.846·24-s + 0.739·25-s + 0.791·26-s + 0.357·27-s − 1.12·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.1i) \, \Gamma_{\R}(s+10.4i) \, \Gamma_{\R}(s-17.1i) \, \Gamma_{\R}(s-10.4i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(20.4599\) |
| Root analytic conductor: | \(2.12679\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.12884364388i, 10.4306590599i, -17.12884364388i, -10.4306590599i:\ ),\ 1)\) |
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
−24.43948451, −22.09020755, −19.80841817, −17.74128521, −8.94033760, −6.24681697, −2.24215302, −0.45136761, 0.45136761, 2.24215302, 6.24681697, 8.94033760, 17.74128521, 19.80841817, 22.09020755, 24.43948451