Dirichlet series
| L(s) = 1 | + (−0.679 − 0.434i)2-s + (0.572 + 0.610i)3-s + (−0.128 + 0.591i)4-s + (−0.117 + 0.0119i)5-s + (−0.124 − 0.663i)6-s + (0.953 + 0.364i)7-s + (−0.0628 + 0.263i)8-s + (−0.241 + 0.699i)9-s + (0.0850 + 0.0428i)10-s + (−0.524 + 0.247i)11-s + (−0.434 + 0.260i)12-s + (−0.317 − 0.138i)13-s + (−0.489 − 0.662i)14-s + (−0.0745 − 0.0647i)15-s + (−0.140 − 0.388i)16-s + (0.596 − 0.309i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-23.5i) \, \Gamma_{\R}(s+0.822i) \, \Gamma_{\R}(s+8.12i) \, \Gamma_{\R}(s+14.5i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.07364\) |
| Root analytic conductor: | \(1.01792\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | no |
| Selberg data: | \((4,\ 1,\ (-23.5430066352i, 0.822180974528i, 8.12585455838i, 14.59497110224i:\ ),\ 1)\) |
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
−24.38990213, −23.44783889, −21.05750458, −19.27632994, −17.83304504, 2.60944945, 4.86505332, 7.97719801, 8.97047993, 10.51519076, 11.99956019, 14.07956742, 15.43215124, 17.26088332, 18.54570547, 20.26415920, 21.37532528