Dirichlet series
| L(s) = 1 | + (−0.421 − 1.06i)2-s + (−0.768 + 1.31i)3-s + (−0.541 − 0.167i)4-s + (−0.400 − 0.239i)5-s + (1.72 + 0.266i)6-s + (−0.117 − 0.553i)7-s + (−0.268 + 0.648i)8-s + (−0.366 − 0.703i)9-s + (−0.0872 + 0.528i)10-s + (−0.0411 + 0.100i)11-s + (0.635 − 0.582i)12-s + (−0.309 + 0.328i)13-s + (−0.541 + 0.358i)14-s + (0.622 − 0.341i)15-s + (−0.0226 − 0.546i)16-s + (0.259 + 0.620i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-16.4i) \, \Gamma_{\R}(s-0.171i) \, \Gamma_{\R}(s+16.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(3\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0488655\) |
| Root analytic conductor: | \(0.365595\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((3,\ 1,\ (-16.403124740291375i, -0.17112189172831185i, 16.574246632019687i:\ ),\ 1)\) |
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
−24.2616362328567, −22.6875461169111, −18.3920792539766, −12.3429586556897, −7.8655276953699, −6.4222353306131, 4.6144521141879, 9.8664332915609, 11.1407921358216, 19.8646960903853, 21.6868858848525, 23.4129437373605