Dirichlet series
| L(s) = 1 | + (−1.18 − 0.259i)2-s + (−0.937 − 0.452i)3-s + (0.944 + 0.612i)4-s + (−0.0116 + 0.184i)5-s + (0.989 + 0.777i)6-s + (0.202 + 0.342i)7-s + (−1.68 − 0.609i)8-s + (0.802 + 0.849i)9-s + (0.0616 − 0.214i)10-s + (0.165 + 0.313i)11-s + (−0.608 − 1.00i)12-s + (−0.293 − 0.195i)13-s + (−0.150 − 0.457i)14-s + (0.0944 − 0.167i)15-s + (1.93 + 0.923i)16-s + (−0.155 + 0.605i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.3i) \, \Gamma_{\R}(s-1.78i) \, \Gamma_{\R}(s+6.25i) \, \Gamma_{\R}(s+18.8i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.06472\) |
| Root analytic conductor: | \(1.32311\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-23.3467915216i, -1.78026796581i, 6.25689320186i, 18.87016628548i:\ ),\ 1)\) |
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
−24.69052130, −23.55793043, −21.29614712, −17.36143583, −15.53523224, −11.57051808, −9.59172868, −0.43794484, 6.20232148, 8.01521872, 9.87251034, 11.46460632, 12.51530713, 15.31800018, 17.07002502, 17.97993013, 19.12128902, 21.30368127, 24.86004591