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
−24.456683572, −22.737784517, −21.360091279, −20.319738275, −15.511258986, −11.751809942, −9.696292750, 6.485730757, 7.605965466, 10.906249272, 12.483164499, 14.144390623, 16.531522798, 18.570212028, 22.623360609, 24.118722660, 24.888105826