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Properties

Label 1-571-571.111-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.422 - 0.906i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (−0.789 + 0.614i)2-s + (−0.863 − 0.504i)3-s + (0.245 − 0.969i)4-s + (−0.973 − 0.229i)5-s + (0.991 − 0.131i)6-s + (0.956 − 0.293i)7-s + (0.401 + 0.915i)8-s + (0.490 + 0.871i)9-s + (0.909 − 0.416i)10-s + (−0.909 − 0.416i)11-s + (−0.701 + 0.712i)12-s + (−0.627 − 0.778i)13-s + (−0.574 + 0.818i)14-s + (0.724 + 0.689i)15-s + (−0.879 − 0.475i)16-s + (−0.180 + 0.983i)17-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)2-s + (−0.863 − 0.504i)3-s + (0.245 − 0.969i)4-s + (−0.973 − 0.229i)5-s + (0.991 − 0.131i)6-s + (0.956 − 0.293i)7-s + (0.401 + 0.915i)8-s + (0.490 + 0.871i)9-s + (0.909 − 0.416i)10-s + (−0.909 − 0.416i)11-s + (−0.701 + 0.712i)12-s + (−0.627 − 0.778i)13-s + (−0.574 + 0.818i)14-s + (0.724 + 0.689i)15-s + (−0.879 − 0.475i)16-s + (−0.180 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2951350535 - 0.4630242087i\)
\(L(\frac12)\) \(\approx\) \(0.2951350535 - 0.4630242087i\)
\(L(1)\) \(\approx\) \(0.4945214023 - 0.06921719498i\)
\(L(1)\) \(\approx\) \(0.4945214023 - 0.06921719498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.789 + 0.614i)T \)
3 \( 1 + (-0.863 - 0.504i)T \)
5 \( 1 + (-0.973 - 0.229i)T \)
7 \( 1 + (0.956 - 0.293i)T \)
11 \( 1 + (-0.909 - 0.416i)T \)
13 \( 1 + (-0.627 - 0.778i)T \)
17 \( 1 + (-0.180 + 0.983i)T \)
19 \( 1 + (0.995 - 0.0990i)T \)
23 \( 1 + (-0.213 - 0.976i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.965 - 0.261i)T \)
41 \( 1 + (0.0825 + 0.996i)T \)
43 \( 1 + (0.115 + 0.993i)T \)
47 \( 1 + (0.879 + 0.475i)T \)
53 \( 1 + (0.340 - 0.940i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.277 + 0.960i)T \)
67 \( 1 + (0.340 - 0.940i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.965 + 0.261i)T \)
79 \( 1 + (0.768 + 0.639i)T \)
83 \( 1 + (0.991 - 0.131i)T \)
89 \( 1 + (-0.991 + 0.131i)T \)
97 \( 1 + (-0.846 - 0.533i)T \)
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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

−23.364679638967103168959054861178, −22.12007145427074227850131276412, −21.74665680904114078490645866664, −20.61183995309992427874183108861, −20.20420534008073760620134290603, −18.923140473021278757134548935706, −18.21232074734931515774625572390, −17.717461356683564770482879965336, −16.62347252704066551788677256437, −15.88660499607806921644510538617, −15.282535511028539855812270750095, −14.025128006571517376540181666907, −12.485886819133406914833742732, −11.88096186695950410420140198608, −11.32280593468415849029165892785, −10.57912083830726126389041122254, −9.61434346719316495613507225181, −8.71819046141683952074327033552, −7.42823828862412712364510820720, −7.17129784669644848280558853114, −5.31902640650474381567375547870, −4.56755089895559209539664167466, −3.522434362978007463559346180376, −2.2868637298423209830826381920, −0.89637943769925853279052904510, 0.310649284970023747900747422757, 1.04359577892569989383836791373, 2.44521393173828496246909666433, 4.399316645587022050747657402466, 5.18721211657563656644192728148, 6.04505775494107557862365446293, 7.30683728110711302903203881342, 7.87974735914220549317402156328, 8.35562785876855045627285825610, 9.96192465693780285163968139158, 10.86340115338750785182069477205, 11.33482186878557749306611872001, 12.3890933841411365101738464849, 13.37969121600674028644213285775, 14.62590528664377004450970652993, 15.372618417011722608558879030590, 16.29968067316942226138613355483, 16.85921082904709916796330146793, 17.91541105118807989380525404370, 18.24006811289400821242010123171, 19.30537466132364449800492533585, 19.974162361677971863870904711384, 20.93519532414431665262560543719, 22.25306517786273114923784552204, 23.14847556366895127849899147370

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