Dirichlet series
| L(s) = 1 | + (0.249 − 0.0197i)2-s + (−1.10 − 0.271i)3-s + (−0.597 − 0.00986i)4-s + (0.483 + 0.742i)5-s + (−0.280 − 0.0459i)6-s + (−0.0831 − 0.372i)7-s + (−0.0644 + 0.0421i)8-s + (0.446 + 0.597i)9-s + (0.135 + 0.176i)10-s + (−0.844 − 0.256i)11-s + (0.655 + 0.172i)12-s + (1.01 + 0.0905i)13-s + (−0.0281 − 0.0912i)14-s + (−0.331 − 0.950i)15-s + (−0.558 + 0.0183i)16-s + (0.586 − 0.138i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+22.5i) \, \Gamma_{\R}(s+0.257i) \, \Gamma_{\R}(s-5.88i) \, \Gamma_{\R}(s-16.9i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0983840\) |
| Root analytic conductor: | \(0.560055\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (22.5466403054i, 0.257932398066i, -5.88708690456i, -16.91748579892i:\ ),\ 1)\) |
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
−23.41421926, −21.04538032, −18.30570202, −17.10197137, −15.69069147, −13.49610449, −12.26577818, −10.49168238, −8.79194008, −5.98568168, −4.70854054, 10.82988728, 13.45358330, 18.44449218, 21.44845138, 22.90452478, 23.40433135