Properties

Degree 1
Conductor 569
Sign $-0.990 - 0.139i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.240 − 0.970i)2-s + (−0.894 − 0.448i)3-s + (−0.883 + 0.467i)4-s + (0.759 − 0.650i)5-s + (−0.219 + 0.975i)6-s + (−0.862 + 0.506i)7-s + (0.666 + 0.745i)8-s + (0.598 + 0.801i)9-s + (−0.814 − 0.580i)10-s + (−0.544 + 0.839i)11-s + (0.999 − 0.0221i)12-s + (0.367 − 0.930i)13-s + (0.699 + 0.714i)14-s + (−0.970 + 0.240i)15-s + (0.562 − 0.826i)16-s + (0.197 + 0.980i)17-s + ⋯
L(s,χ)  = 1  + (−0.240 − 0.970i)2-s + (−0.894 − 0.448i)3-s + (−0.883 + 0.467i)4-s + (0.759 − 0.650i)5-s + (−0.219 + 0.975i)6-s + (−0.862 + 0.506i)7-s + (0.666 + 0.745i)8-s + (0.598 + 0.801i)9-s + (−0.814 − 0.580i)10-s + (−0.544 + 0.839i)11-s + (0.999 − 0.0221i)12-s + (0.367 − 0.930i)13-s + (0.699 + 0.714i)14-s + (−0.970 + 0.240i)15-s + (0.562 − 0.826i)16-s + (0.197 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.990 - 0.139i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.990 - 0.139i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.990 - 0.139i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (85, \cdot )$
Sato-Tate  :  $\mu(284)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.990 - 0.139i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.04066842227 - 0.5800731235i$
$L(\frac12,\chi)$  $\approx$  $0.04066842227 - 0.5800731235i$
$L(\chi,1)$  $\approx$  0.4839351548 - 0.4095281273i
$L(1,\chi)$  $\approx$  0.4839351548 - 0.4095281273i

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

−23.445335174204384825007441406341, −23.02642402201321303972959527431, −22.09583324534186570858045326904, −21.61566344076752749829760339301, −20.453875013665464692313609279457, −18.93026490609457990152790180781, −18.50174965870082064484591800478, −17.7071790417109918723664894508, −16.70067505680556415252195898131, −16.25262035608752203638610489131, −15.61994777956083495368103360405, −14.2183912820370184769775003586, −13.83463316814784518420679095899, −12.798046069044644606124479641413, −11.4750745045271619031652047908, −10.46186584283215697081983082387, −9.85488282969293759789993030188, −9.17866747190258066511169756754, −7.71584708111569900496571161164, −6.701240153730993199125967830179, −6.14641357050448000901540509034, −5.42715637037244711127356296588, −4.22556242910073213305676969044, −3.170877351059313937756453821291, −1.18802354944889895656649325509, 0.419553036268576800579409348874, 1.72135358410029869283323942296, 2.55348474955776116139504337527, 3.99425789433006262009954547657, 5.27370266508196356290540222431, 5.710284930621160136587379867150, 7.05163921432279335074676253863, 8.245020968248200066770389929159, 9.19450419269850289109953288608, 10.23276229837794689479514907303, 10.55672866498698820197872681985, 12.01762087448757292184317558338, 12.56445829820126147880519243783, 13.04199076929547197651900153763, 13.874263836979508085346131685937, 15.52832020463927825066154898455, 16.36001651615209606150670214618, 17.40474206439295870655174785767, 17.85448716925116679617237704464, 18.53121124179116552281914498196, 19.6269224459242832654328800628, 20.24930119572044794043482266363, 21.35105217712983748349845105713, 21.969228831566525395596443235340, 22.65516313138333054985703752898

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