Dirichlet series
| L(s) = 1 | + (0.0582 − 0.197i)2-s + (−0.382 + 1.42i)3-s + (0.573 − 0.0229i)4-s + (−0.129 − 0.576i)5-s + (0.259 + 0.158i)6-s + (−0.00648 + 0.436i)7-s + (0.122 − 0.0372i)8-s + (−0.664 − 1.09i)9-s + (−0.121 − 0.00794i)10-s + (0.308 + 0.472i)11-s + (−0.186 + 0.826i)12-s + (−0.238 − 0.170i)13-s + (0.0858 + 0.0267i)14-s + (0.871 + 0.0350i)15-s + (−0.609 − 0.0403i)16-s + (0.792 − 0.305i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+19.9i) \, \Gamma_{\R}(s+1.68i) \, \Gamma_{\R}(s-4.66i) \, \Gamma_{\R}(s-16.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.64447\) |
| Root analytic conductor: | \(1.13241\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (19.91852255416i, 1.685122031456i, -4.66217749248i, -16.94146709314i:\ ),\ 1)\) |
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
−24.888105826, −24.118722660, −22.623360609, −18.570212028, −16.531522798, −14.144390623, −12.483164499, −10.906249272, −7.605965466, −6.485730757, 9.696292750, 11.751809942, 15.511258986, 20.319738275, 21.360091279, 22.737784517, 24.456683572