Properties

Degree 1
Conductor 569
Sign $-0.646 + 0.762i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.408 − 0.912i)2-s + (−0.996 + 0.0883i)3-s + (−0.666 − 0.745i)4-s + (0.487 − 0.873i)5-s + (−0.325 + 0.945i)6-s + (0.699 − 0.714i)7-s + (−0.952 + 0.304i)8-s + (0.984 − 0.176i)9-s + (−0.598 − 0.801i)10-s + (−0.759 + 0.650i)11-s + (0.730 + 0.683i)12-s + (−0.975 + 0.219i)13-s + (−0.367 − 0.930i)14-s + (−0.408 + 0.912i)15-s + (−0.110 + 0.993i)16-s + (−0.883 − 0.467i)17-s + ⋯
L(s,χ)  = 1  + (0.408 − 0.912i)2-s + (−0.996 + 0.0883i)3-s + (−0.666 − 0.745i)4-s + (0.487 − 0.873i)5-s + (−0.325 + 0.945i)6-s + (0.699 − 0.714i)7-s + (−0.952 + 0.304i)8-s + (0.984 − 0.176i)9-s + (−0.598 − 0.801i)10-s + (−0.759 + 0.650i)11-s + (0.730 + 0.683i)12-s + (−0.975 + 0.219i)13-s + (−0.367 − 0.930i)14-s + (−0.408 + 0.912i)15-s + (−0.110 + 0.993i)16-s + (−0.883 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.646 + 0.762i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.646 + 0.762i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.646 + 0.762i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (130, \cdot )$
Sato-Tate  :  $\mu(142)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.646 + 0.762i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2695092579 - 0.5820902851i$
$L(\frac12,\chi)$  $\approx$  $-0.2695092579 - 0.5820902851i$
$L(\chi,1)$  $\approx$  0.5262921534 - 0.5997052308i
$L(1,\chi)$  $\approx$  0.5262921534 - 0.5997052308i

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

−23.98499947312996829071096933949, −22.89947830934383735379325592207, −22.17692865386042168783550423122, −21.68459912608918482433102138212, −21.063448346306726692710236171813, −19.2528729585828904168632746904, −18.273887405077708299674113341626, −17.84774907755957744364521776974, −17.257223200903261831308320989816, −16.04096778105291219293820765757, −15.52031430137156825355296924584, −14.51281117664730124256473215393, −13.81727948096192436415251382943, −12.72257264127088798679077728336, −11.985221895935746682625527377126, −10.987111966702658511910246731779, −10.12827855080166021670910939346, −8.94100415782856349343205358179, −7.72190314896657084508438323564, −7.10214120101466348021649623291, −5.840084159190431350463307055116, −5.61919138347223409817073224626, −4.58902168265345991294655099388, −3.210518556943339697019842429810, −1.95302839987612363894721179561, 0.322441353163615287649784052868, 1.5128405244797035092402857413, 2.50185994362846515703442436360, 4.506721502723677427839877034058, 4.58391610991455674854428543171, 5.45910389396299846906174045460, 6.678851513778197381919381877863, 7.87686431286931572768158887966, 9.29165200920569373567468590775, 9.980351784270659744289190252402, 10.75709195323096508084887128883, 11.67075585510846952995692496589, 12.36827839862754853344058232655, 13.1958560259461144866892422477, 13.88911281785115659055758805431, 15.06251020478883504665302141080, 16.076816898136639865795585743707, 17.12195674142882455419313111973, 17.793487697662309897183608530130, 18.280389755978208962991585169247, 19.76851723911495272029125822253, 20.37036970707490960254907517240, 21.09949218166496924409597173429, 21.85180446420425464803113784715, 22.59319333336867176051488952506

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