Dirichlet series
L(s) = 1 | + (0.674 + 0.590i)2-s + (−0.382 + 0.156i)3-s + (0.551 + 0.796i)4-s + (−0.535 + 0.125i)5-s + (−0.350 − 0.119i)6-s + (0.151 − 0.120i)7-s + (0.876 + 0.535i)8-s + (0.569 − 0.119i)9-s + (−0.435 − 0.231i)10-s + (−0.602 − 0.457i)11-s + (−0.335 − 0.217i)12-s + (−0.602 − 0.00348i)13-s + (0.173 + 0.00785i)14-s + (0.184 − 0.131i)15-s + (0.324 + 1.23i)16-s + (0.290 + 0.468i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.2i) \, \Gamma_{\R}(s+1.85i) \, \Gamma_{\R}(s-4.50i) \, \Gamma_{\R}(s-19.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
Degree: | \(4\) |
Conductor: | \(1\) |
Sign: | $1$ |
Analytic conductor: | \(2.27924\) |
Root analytic conductor: | \(1.22870\) |
Rational: | no |
Arithmetic: | no |
Primitive: | yes |
Self-dual: | no |
Selberg data: | \((4,\ 1,\ (22.2500458752i, 1.852039992048i, -4.50889918758i, -19.59318667958i:\ ),\ 1)\) |
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
−23.39273141, −20.44072297, −18.83244589, −16.48950296, −14.87732783, −13.00557715, −11.66655489, −10.32196893, −7.35350569, −4.90148759, 7.39147020, 11.01853893, 13.02429540, 15.24744890, 16.66008845, 21.76685994, 23.30359953, 24.48310810