Properties

Label 1-571-571.220-r0-0-0
Degree $1$
Conductor $571$
Sign $0.922 - 0.385i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.904 − 0.426i)2-s + (0.716 + 0.697i)3-s + (0.635 − 0.771i)4-s + (0.451 − 0.892i)5-s + (0.945 + 0.324i)6-s + (−0.677 + 0.735i)7-s + (0.245 − 0.969i)8-s + (0.0275 + 0.999i)9-s + (0.0275 − 0.999i)10-s + (0.851 − 0.523i)11-s + (0.993 − 0.110i)12-s + (0.137 + 0.990i)13-s + (−0.298 + 0.954i)14-s + (0.945 − 0.324i)15-s + (−0.191 − 0.981i)16-s + (−0.191 − 0.981i)17-s + ⋯
L(s)  = 1  + (0.904 − 0.426i)2-s + (0.716 + 0.697i)3-s + (0.635 − 0.771i)4-s + (0.451 − 0.892i)5-s + (0.945 + 0.324i)6-s + (−0.677 + 0.735i)7-s + (0.245 − 0.969i)8-s + (0.0275 + 0.999i)9-s + (0.0275 − 0.999i)10-s + (0.851 − 0.523i)11-s + (0.993 − 0.110i)12-s + (0.137 + 0.990i)13-s + (−0.298 + 0.954i)14-s + (0.945 − 0.324i)15-s + (−0.191 − 0.981i)16-s + (−0.191 − 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.922 - 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.175605446 - 0.6375253173i\)
\(L(\frac12)\) \(\approx\) \(3.175605446 - 0.6375253173i\)
\(L(1)\) \(\approx\) \(2.256936745 - 0.3304294247i\)
\(L(1)\) \(\approx\) \(2.256936745 - 0.3304294247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.904 - 0.426i)T \)
3 \( 1 + (0.716 + 0.697i)T \)
5 \( 1 + (0.451 - 0.892i)T \)
7 \( 1 + (-0.677 + 0.735i)T \)
11 \( 1 + (0.851 - 0.523i)T \)
13 \( 1 + (0.137 + 0.990i)T \)
17 \( 1 + (-0.191 - 0.981i)T \)
19 \( 1 + (0.716 + 0.697i)T \)
23 \( 1 + (0.245 + 0.969i)T \)
29 \( 1 + (-0.926 - 0.376i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (0.137 - 0.990i)T \)
41 \( 1 + (0.451 - 0.892i)T \)
43 \( 1 + (0.0275 - 0.999i)T \)
47 \( 1 + (-0.754 + 0.656i)T \)
53 \( 1 + (0.904 + 0.426i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.821 - 0.569i)T \)
67 \( 1 + (-0.821 + 0.569i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.926 + 0.376i)T \)
79 \( 1 + (0.350 - 0.936i)T \)
83 \( 1 + (-0.754 + 0.656i)T \)
89 \( 1 + (-0.191 - 0.981i)T \)
97 \( 1 + (0.635 - 0.771i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.09642825691534518661665755552, −22.68232954112713332034670406878, −21.900450982226554899369549537353, −20.75855830746891124905302554464, −19.998633502208509526404768550623, −19.415513332595940371051428608227, −18.14254301875861068377859019332, −17.4600734817012531538576303194, −16.57866246195149373462425360673, −15.12283654540329342490742716225, −14.92523668870236917798961889363, −13.89089664425554853282507072577, −13.261997051179372574234741669210, −12.65802130182660975140442465824, −11.52606754180081804927860594351, −10.44854599951606434530770860378, −9.44467509461805359541836532991, −8.160077207710713732249801802770, −7.2824343510248309477633007245, −6.59173431191958399802380793668, −6.0342816202662772546966145256, −4.40971587580685367352063494857, −3.34328595473111887808099101408, −2.81109942775951548156826280353, −1.51191248254768197004314623588, 1.4476088303246715643724621200, 2.46821862469289574960306931159, 3.5108621485954745067330469591, 4.30448758401837819491224610090, 5.36231117404957401261728526744, 6.03649723846260419868077789430, 7.36290780164907076797049753184, 9.04009001687244683283978962350, 9.21367654156375673696474043746, 10.13003993207027285351008344694, 11.49449378421552186397948019655, 12.06210032456478532910978791861, 13.23257935278280195035108951724, 13.82855255725960344609121720082, 14.48614131153843096768139106936, 15.72834279764424848771577076308, 16.08576180249081746474198710585, 16.9435034261042199806176025135, 18.610619914639243329244271838561, 19.37084222311077759060676182422, 20.025405468444555718390290107170, 20.97691048069730349881687830561, 21.38377112297640248191366143488, 22.20121783451732401536289159492, 22.861721107416636772268540807764

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