Properties

Degree 1
Conductor 569
Sign $0.341 - 0.939i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.633 − 0.773i)2-s + (0.952 − 0.304i)3-s + (−0.197 − 0.980i)4-s + (−0.839 + 0.544i)5-s + (0.367 − 0.930i)6-s + (0.937 + 0.346i)7-s + (−0.883 − 0.467i)8-s + (0.814 − 0.580i)9-s + (−0.110 + 0.993i)10-s + (0.787 + 0.616i)11-s + (−0.487 − 0.873i)12-s + (0.699 + 0.714i)13-s + (0.862 − 0.506i)14-s + (−0.633 + 0.773i)15-s + (−0.921 + 0.387i)16-s + (−0.991 − 0.132i)17-s + ⋯
L(s,χ)  = 1  + (0.633 − 0.773i)2-s + (0.952 − 0.304i)3-s + (−0.197 − 0.980i)4-s + (−0.839 + 0.544i)5-s + (0.367 − 0.930i)6-s + (0.937 + 0.346i)7-s + (−0.883 − 0.467i)8-s + (0.814 − 0.580i)9-s + (−0.110 + 0.993i)10-s + (0.787 + 0.616i)11-s + (−0.487 − 0.873i)12-s + (0.699 + 0.714i)13-s + (0.862 − 0.506i)14-s + (−0.633 + 0.773i)15-s + (−0.921 + 0.387i)16-s + (−0.991 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.341 - 0.939i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.341 - 0.939i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.341 - 0.939i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (34, \cdot )$
Sato-Tate  :  $\mu(142)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.341 - 0.939i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.168335731 - 1.518563425i$
$L(\frac12,\chi)$  $\approx$  $2.168335731 - 1.518563425i$
$L(\chi,1)$  $\approx$  1.715317699 - 0.8408371421i
$L(1,\chi)$  $\approx$  1.715317699 - 0.8408371421i

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

−23.61406520428374287440828894788, −22.677297849843746678691848942463, −21.78727301884680329425029020595, −20.70891513076339361745098319587, −20.51164275673574254282597891000, −19.42364850904557980039377545429, −18.39851229747768609967613668186, −17.241254498485493962681423126869, −16.50153746800292633994000505341, −15.5679859625990356016059106572, −15.09077553987538573271476290608, −14.14935915843825699875912291583, −13.52373460288524728613192068577, −12.59287564002219305291000533585, −11.5667854916274414974955791335, −10.6686777873606450592034472209, −8.97602100967542556041038496344, −8.52562881694533316209479403939, −7.82655037147951387523366561286, −6.953178866464136755486551482986, −5.566134978726552758066265483274, −4.508274365578211130610082869876, −3.91756784188649010319147281169, −3.063059553770726130711729768508, −1.37968829221717326142852018251, 1.30541091049805628858250734581, 2.23428170853566245338569841413, 3.21371697098579309123240573685, 4.17999233592722842686876128862, 4.83717828739133918297687882118, 6.62078910719257966368623885453, 7.12665959330374384013755778818, 8.6674770548267108122430479254, 8.99298441509241417231654063372, 10.38564740584792994637199881335, 11.38623446984731824771162857117, 11.84667889451207013945129969955, 12.89937874447129266796256291262, 13.87290312373406346006582664904, 14.50935742788809332014814024778, 15.19179094534332391299215389844, 15.81893306701499394630403308276, 17.73807464706522074388495194702, 18.31029392576877942964793034929, 19.290119296365334338977079436629, 19.73497552567807038868980015144, 20.55898875357442901435842272071, 21.34623597399084100855082424137, 22.12522343589847897297132465233, 23.1358872509353360275244613982

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