Dirichlet series
| L(s) = 1 | + (1.25 + 0.746i)2-s + (0.310 − 1.45i)3-s + (0.400 + 1.87i)4-s + (−0.0767 + 0.0627i)5-s + (1.47 − 1.59i)6-s + (−0.540 + 0.326i)7-s + (−0.419 + 1.44i)8-s + (−0.898 − 0.906i)9-s + (−0.143 + 0.0215i)10-s + (−0.382 − 0.203i)11-s + (2.85 − 0.00108i)12-s + (−0.0268 + 0.182i)13-s + (−0.922 + 0.00613i)14-s + (0.0677 + 0.131i)15-s + (−0.718 + 0.339i)16-s + (0.361 + 0.291i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-21.1i) \, \Gamma_{\R}(s-3.29i) \, \Gamma_{\R}(s+3.44i) \, \Gamma_{\R}(s+21.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.21079\) |
| Root analytic conductor: | \(1.33860\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-21.1709752712i, -3.29346016214i, 3.44054733534i, 21.023888098i:\ ),\ 1)\) |
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
−23.5760440, −22.1860928, −19.8074830, −15.9923107, −14.6742523, −13.2972617, −10.8924504, −9.8163386, −4.9323140, 6.5620540, 8.0103796, 12.4318524, 12.9878376, 14.4237577, 16.4912885, 18.5813021, 22.8439508, 24.2069230