Dirichlet series
| L(s) = 1 | − 0.234·2-s + 1.79·3-s − 1.00·4-s + 0.394·5-s − 0.421·6-s − 0.193·7-s + 0.249·8-s + 0.996·9-s − 0.0924·10-s + 0.225·11-s − 1.80·12-s + 0.237·13-s + 0.0453·14-s + 0.708·15-s + 0.0624·16-s + 0.738·17-s − 0.233·18-s − 0.469·19-s − 0.396·20-s − 0.347·21-s − 0.0529·22-s + 0.984·23-s + 0.448·24-s − 0.856·25-s − 0.0550·26-s − 0.422·27-s + 0.190·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+17.8i) \, \Gamma_{\R}(s+3.19i) \, \Gamma_{\R}(s-17.8i) \, \Gamma_{\R}(s-3.19i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.07877\) |
| Root analytic conductor: | \(1.20074\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (17.88146337722i, 3.19726747374i, -17.88146337722i, -3.19726747374i:\ ),\ 1)\) |
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
−23.019924686, −21.157781506, −19.506552962, −14.513267270, −13.408203181, −9.482159710, −8.402435745, 8.402435745, 9.482159710, 13.408203181, 14.513267270, 19.506552962, 21.157781506, 23.019924686