Properties

Degree 1
Conductor 569
Sign $0.453 + 0.891i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.903 + 0.428i)2-s + (0.154 − 0.988i)3-s + (0.633 + 0.773i)4-s + (−0.283 + 0.958i)5-s + (0.562 − 0.826i)6-s + (0.984 − 0.176i)7-s + (0.240 + 0.970i)8-s + (−0.952 − 0.304i)9-s + (−0.666 + 0.745i)10-s + (0.325 + 0.945i)11-s + (0.862 − 0.506i)12-s + (−0.921 + 0.387i)13-s + (0.964 + 0.262i)14-s + (0.903 + 0.428i)15-s + (−0.197 + 0.980i)16-s + (0.0663 + 0.997i)17-s + ⋯
L(s,χ)  = 1  + (0.903 + 0.428i)2-s + (0.154 − 0.988i)3-s + (0.633 + 0.773i)4-s + (−0.283 + 0.958i)5-s + (0.562 − 0.826i)6-s + (0.984 − 0.176i)7-s + (0.240 + 0.970i)8-s + (−0.952 − 0.304i)9-s + (−0.666 + 0.745i)10-s + (0.325 + 0.945i)11-s + (0.862 − 0.506i)12-s + (−0.921 + 0.387i)13-s + (0.964 + 0.262i)14-s + (0.903 + 0.428i)15-s + (−0.197 + 0.980i)16-s + (0.0663 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.453 + 0.891i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.453 + 0.891i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.453 + 0.891i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (113, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.453 + 0.891i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.125499624 + 1.302519269i$
$L(\frac12,\chi)$  $\approx$  $2.125499624 + 1.302519269i$
$L(\chi,1)$  $\approx$  1.773676722 + 0.5146428453i
$L(1,\chi)$  $\approx$  1.773676722 + 0.5146428453i

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

−22.954493932564545453050919331816, −22.04207191241934120812716855283, −21.51884329230614811667403906957, −20.48094398302026108732162458127, −20.37268405293390277161773802956, −19.36137890028990622717870738629, −18.17730482216581576007530190673, −16.77936058698756957549545780287, −16.30196583611996770977788412373, −15.443024311485810167772622088827, −14.368948262396333877295515625982, −14.16131798656354338297473944810, −12.79030799205832556636686215024, −11.89457228911224052293174938540, −11.30274038943861898251969977798, −10.35980571055168535997832473468, −9.34473909303085844811407339477, −8.50026976984044393610344848453, −7.42081628248708547450604595195, −5.69844989808841466832415802217, −5.164165652561199160161856622218, −4.44332067667299237537948787433, −3.48840334556118468353308120278, −2.42077664997130564435709471876, −0.97000044925117110740371886318, 1.78432361998636946039269945952, 2.48939869216517514578953763883, 3.71378400679073279615206914363, 4.726506503710701374000664832052, 5.890790451598677099671884216433, 6.83643690156846644589839006952, 7.546339914257380491530545746050, 7.960204102960062379348431682001, 9.50044706048605881632976073189, 11.044142588077808080397568519170, 11.60296587525555767093226570509, 12.35266887735857792898703985173, 13.362453552597777862607964492869, 14.19979606270493360159747732837, 14.8240801414144757476017162024, 15.29158786800407437954833839186, 16.877160815166180785633881645732, 17.57945902092571642857911523606, 18.14055154099462441934628556735, 19.517650173442999204359624613649, 19.90136533556749657697614009396, 21.08367629350399472082348691991, 21.97523192467243824122299466518, 22.71310458571851687071687751295, 23.58166364049735346487506224368

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