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
−24.5534336, −22.6529068, −21.3569065, −19.6138585, −18.2098176, −16.3237089, −15.6116312, −13.8782684, −11.2125535, −10.7556986, −8.4921964, −7.1390007, −5.0567770, −2.8691006, 1.6692601, 4.6552568, 7.0393962, 8.2284807, 10.3200747, 11.8748201, 13.0537522, 15.2190200, 16.8444509, 18.1568941, 19.8837066, 20.5900876, 23.2172276, 24.2549792