Properties

Degree 1
Conductor 569
Sign $0.892 - 0.451i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0442 − 0.999i)2-s + (0.294 − 0.955i)3-s + (−0.996 + 0.0883i)4-s + (0.903 + 0.428i)5-s + (−0.967 − 0.251i)6-s + (0.945 + 0.325i)7-s + (0.132 + 0.991i)8-s + (−0.826 − 0.562i)9-s + (0.387 − 0.921i)10-s + (0.999 + 0.0331i)11-s + (−0.208 + 0.977i)12-s + (−0.346 + 0.937i)13-s + (0.283 − 0.958i)14-s + (0.675 − 0.737i)15-s + (0.984 − 0.176i)16-s + (0.894 − 0.448i)17-s + ⋯
L(s,χ)  = 1  + (−0.0442 − 0.999i)2-s + (0.294 − 0.955i)3-s + (−0.996 + 0.0883i)4-s + (0.903 + 0.428i)5-s + (−0.967 − 0.251i)6-s + (0.945 + 0.325i)7-s + (0.132 + 0.991i)8-s + (−0.826 − 0.562i)9-s + (0.387 − 0.921i)10-s + (0.999 + 0.0331i)11-s + (−0.208 + 0.977i)12-s + (−0.346 + 0.937i)13-s + (0.283 − 0.958i)14-s + (0.675 − 0.737i)15-s + (0.984 − 0.176i)16-s + (0.894 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.892 - 0.451i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.892 - 0.451i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.892 - 0.451i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (12, \cdot )$
Sato-Tate  :  $\mu(568)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (1:\ ),\ 0.892 - 0.451i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.386325778 - 0.5694720885i$
$L(\frac12,\chi)$  $\approx$  $2.386325778 - 0.5694720885i$
$L(\chi,1)$  $\approx$  1.201184623 - 0.6059389047i
$L(1,\chi)$  $\approx$  1.201184623 - 0.6059389047i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−23.05771065244905902964001020381, −22.23654046774586954526308705209, −21.59160994219434183275671628724, −20.76633706556035709597994200140, −19.95428416061892636518653131407, −18.81123541363800442378341029311, −17.43145159516566356699326772968, −17.2896111260052161457462896301, −16.5444468725568389281728266494, −15.41323904366803312073482046346, −14.64926214926677016773740594426, −14.16724484992687324206428609440, −13.28802710340295413694387486794, −12.12967620826884979030159046362, −10.65507116172109387541211143054, −10.02560465798622051238215266690, −9.07653918250935121767345930979, −8.430448898312383764240782529663, −7.52302999636454249439056809038, −6.102055271624785450572561309693, −5.41806205933103949923431459665, −4.53687805784573247729289163967, −3.74370113584724166205069678508, −2.068382991906817228780700190500, −0.593862924796432584446370137580, 1.37903159123816143985003585543, 1.737536857329454288634741622, 2.73963817793250110888700238772, 3.90897491704965528473254993438, 5.233331180919102046674515663293, 6.1800175841015018776391030296, 7.31156076187146211023565190530, 8.389538097053551494207795568010, 9.1897572975186432199023397334, 9.99516412060680843940455727373, 11.24369990113722470400143604325, 11.86112811668652865532945922843, 12.591573113435183926189779560, 13.7225762645313672998407090456, 14.37171470011167063985032779865, 14.61927279349254406591027121455, 16.79837503139531680831057593701, 17.47471152833391332092326645100, 18.13806844282990403161257238967, 18.90502208030661174896947616687, 19.45728516046162510412553468229, 20.62180417014084813127538206828, 21.18527705393407251038392886285, 21.99507367637540165705470300493, 22.84982930820819966806032675843

Graph of the $Z$-function along the critical line