Properties

Degree 1
Conductor 569
Sign $-0.754 - 0.655i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.132 + 0.991i)2-s + (−0.780 + 0.624i)3-s + (−0.964 + 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.722 − 0.691i)6-s + (0.544 + 0.839i)7-s + (−0.387 − 0.921i)8-s + (0.219 − 0.975i)9-s + (−0.930 + 0.367i)10-s + (0.995 + 0.0993i)11-s + (0.589 − 0.807i)12-s + (0.873 − 0.487i)13-s + (−0.759 + 0.650i)14-s + (−0.794 − 0.607i)15-s + (0.862 − 0.506i)16-s + (0.176 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (0.132 + 0.991i)2-s + (−0.780 + 0.624i)3-s + (−0.964 + 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.722 − 0.691i)6-s + (0.544 + 0.839i)7-s + (−0.387 − 0.921i)8-s + (0.219 − 0.975i)9-s + (−0.930 + 0.367i)10-s + (0.995 + 0.0993i)11-s + (0.589 − 0.807i)12-s + (0.873 − 0.487i)13-s + (−0.759 + 0.650i)14-s + (−0.794 − 0.607i)15-s + (0.862 − 0.506i)16-s + (0.176 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.754 - 0.655i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.754 - 0.655i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.754 - 0.655i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (21, \cdot )$
Sato-Tate  :  $\mu(568)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (1:\ ),\ -0.754 - 0.655i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.5823793137 + 1.558091700i$
$L(\frac12,\chi)$  $\approx$  $-0.5823793137 + 1.558091700i$
$L(\chi,1)$  $\approx$  0.5093998253 + 0.8615541753i
$L(1,\chi)$  $\approx$  0.5093998253 + 0.8615541753i

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

−22.52951767394607589839919345507, −21.62189687414276540256263099219, −20.889156481379447807592278912049, −20.02747748129612454948493238651, −19.36058723637686899708034494424, −18.39695580092094075438857427985, −17.412786658341943493305395761246, −17.0941750292014112664233924519, −16.14181425647865246255595453516, −14.459664214904857366821966777417, −13.74334752651322613279564996968, −12.9577201020593125894816406109, −12.33859348408013661600578021840, −11.21658947555876176182443560746, −10.975831283513201128749106898246, −9.64349931592989523800262715798, −8.75446524517116463014225518255, −7.814761782892395988138077648930, −6.4626397026668198062840877880, −5.53338566688900524152046435445, −4.50423100629764117567266904150, −3.879509235723310019537160689137, −1.99647197023088395472403801115, −1.24320316996742406976774078360, −0.53046982620686096019243007423, 1.26500710977109344131227452329, 3.21032417690021382160907156920, 3.9997779366519053413018332936, 5.413796883898186241112364466220, 5.71654068722344502452012511120, 6.74764288696930122863847883167, 7.60248135102033846331286849626, 8.98213782940239254163763136920, 9.529599170479419093322065465467, 10.68871991819040957923067351257, 11.584309797878073150070122774550, 12.368163245935181387340583644278, 13.7428103624879915798630412364, 14.53060308411926403260841693052, 15.192986123780573274995711110952, 15.89356362993706362778878626704, 16.7881433381139743721166201200, 17.72366002778605519874355019002, 18.214654702299403812273648306017, 18.88227324435913518693953773854, 20.56728174552941767661690001715, 21.442604122831570248752940544566, 22.27767307885648926733481072788, 22.572787835521916420719948757026, 23.42050682988557044066131627478

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