Properties

Degree 1
Conductor 569
Sign $-0.851 + 0.524i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.759 − 0.650i)2-s + (−0.997 + 0.0663i)3-s + (0.154 + 0.988i)4-s + (0.699 − 0.714i)5-s + (0.801 + 0.598i)6-s + (−0.562 − 0.826i)7-s + (0.525 − 0.850i)8-s + (0.991 − 0.132i)9-s + (−0.996 + 0.0883i)10-s + (−0.506 − 0.862i)11-s + (−0.219 − 0.975i)12-s + (−0.814 − 0.580i)13-s + (−0.110 + 0.993i)14-s + (−0.650 + 0.759i)15-s + (−0.952 + 0.304i)16-s + (−0.408 − 0.912i)17-s + ⋯
L(s,χ)  = 1  + (−0.759 − 0.650i)2-s + (−0.997 + 0.0663i)3-s + (0.154 + 0.988i)4-s + (0.699 − 0.714i)5-s + (0.801 + 0.598i)6-s + (−0.562 − 0.826i)7-s + (0.525 − 0.850i)8-s + (0.991 − 0.132i)9-s + (−0.996 + 0.0883i)10-s + (−0.506 − 0.862i)11-s + (−0.219 − 0.975i)12-s + (−0.814 − 0.580i)13-s + (−0.110 + 0.993i)14-s + (−0.650 + 0.759i)15-s + (−0.952 + 0.304i)16-s + (−0.408 − 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.851 + 0.524i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.851 + 0.524i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.851 + 0.524i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (32, \cdot )$
Sato-Tate  :  $\mu(284)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.851 + 0.524i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.09813733152 - 0.3467130117i$
$L(\frac12,\chi)$  $\approx$  $-0.09813733152 - 0.3467130117i$
$L(\chi,1)$  $\approx$  0.3894029906 - 0.3054115256i
$L(1,\chi)$  $\approx$  0.3894029906 - 0.3054115256i

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

−23.92735605936603074653187959318, −22.8414242531726131133646723414, −22.26906325574069734911051872755, −21.58250956773569033093146989004, −20.27083936994728124312269611776, −19.11832302701866599843270640110, −18.44947681120065516892578098297, −17.938866006738469617989064774753, −17.11973966568979305177173991270, −16.3898360912124231381617718399, −15.359716514242621901857571800429, −14.88398662911865694859907503875, −13.63365028304275808898593128904, −12.58664069546074207409038193503, −11.626170902515050283223254878422, −10.62326610806267862292463501548, −9.8438586532548854589543744488, −9.3775696227286946802099715028, −7.861767433369025801078776160655, −6.91013466529383149207199694924, −6.29048152891907732152133111222, −5.52951188590425506088092070437, −4.58168869431222377215332606850, −2.55138316492076883388406117817, −1.69025680549797277112875232092, 0.29970542735014409997592085284, 1.180355643364916831349473548795, 2.61029030415532966640374579213, 3.849342572202543210455737490, 5.00328868105485471314476717306, 5.91578712786416424081695431258, 7.125254667602945052006149851855, 7.88957246710443477093142952983, 9.32714539920775804198496467840, 9.852532867592700749925909724422, 10.59839669222259472233855028375, 11.52868263521938454426370991281, 12.37420517965292011421895637685, 13.21961061706837297180777507583, 13.79029981361501612064395355145, 15.97508746760592937441366648249, 16.09658115366904844551112066251, 17.21798589821749248168833836475, 17.537395892265600264982522682380, 18.47226895036512128041186650184, 19.41718213901673008437384257876, 20.40937945364315128435726376765, 20.91051314970724789644032627566, 22.10414616797183387896208856098, 22.308568361413359954222568810711

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