Dirichlet series
| L(s) = 1 | + (−0.435 + 0.244i)2-s + (−0.155 + 0.696i)3-s + (0.564 + 0.0318i)4-s + (−0.561 + 0.854i)5-s + (−0.103 − 0.340i)6-s + (0.552 + 0.386i)7-s + (0.497 + 0.124i)8-s + (−0.305 + 0.480i)9-s + (0.0349 − 0.509i)10-s + (−0.901 − 1.46i)11-s + (−0.109 + 0.387i)12-s + (0.0339 − 0.0158i)13-s + (−0.334 − 0.0328i)14-s + (−0.507 − 0.523i)15-s + (−0.444 + 0.464i)16-s + (−0.738 − 1.15i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+33.7i) \, \Gamma_{\R}(s+0.275i) \, \Gamma_{\R}(s-33.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.345607\) |
| Root analytic conductor: | \(0.701769\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (33.7130713748i, 0.275047423258i, -33.988118798i:\ ),\ 1)\) |
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
−24.2549792, −23.2172276, −20.5900876, −19.8837066, −18.1568941, −16.8444509, −15.2190200, −13.0537522, −11.8748201, −10.3200747, −8.2284807, −7.0393962, −4.6552568, −1.6692601, 2.8691006, 5.0567770, 7.1390007, 8.4921964, 10.7556986, 11.2125535, 13.8782684, 15.6116312, 16.3237089, 18.2098176, 19.6138585, 21.3569065, 22.6529068, 24.5534336