Dirichlet series
| L(s) = 1 | + (−0.250 + 0.433i)2-s + (0.360 − 0.356i)3-s + (−0.125 − 0.216i)4-s + (−0.592 − 0.583i)5-s + (0.0642 + 0.245i)6-s + (0.103 − 1.02i)7-s +(0.125)·8-s + (−0.357 − 0.613i)9-s + (0.400 − 0.110i)10-s + (1.34 − 0.122i)11-s + (−0.122 − 0.0334i)12-s + (0.0616 + 0.416i)13-s + (0.418 + 0.301i)14-s + (−0.421 + 0.000934i)15-s + (−0.0312 + 0.0541i)16-s + (−0.855 + 0.598i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+8.95i) \, \Gamma_{\R}(s+2.93i) \, \Gamma_{\R}(s-11.8i) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $-0.5 - 0.866i$ |
| Analytic conductor: | \(5.01196\) |
| Root analytic conductor: | \(1.71133\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 4,\ (8.9546625172i, 2.9365915306i, -11.8912540478i:\ ),\ -0.5 - 0.866i)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.59912256, −22.57913504, −22.01745593, −20.29133986, −19.31006000, −17.95238412, −15.92737505, −14.40713480, −11.70886651, 0.93372346, 4.07774542, 6.85564032, 8.72396898, 14.26571491, 16.34014697, 17.69700217, 19.64960226, 20.25256015, 22.34994240, 23.85482092, 24.35540600