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
−24.2069230, −22.8439508, −18.5813021, −16.4912885, −14.4237577, −12.9878376, −12.4318524, −8.0103796, −6.5620540, 4.9323140, 9.8163386, 10.8924504, 13.2972617, 14.6742523, 15.9923107, 19.8074830, 22.1860928, 23.5760440