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Properties

Label 1-571-571.252-r1-0-0
Degree $1$
Conductor $571$
Sign $0.691 + 0.722i$
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  + (0.986 − 0.164i)2-s + (−0.431 + 0.901i)3-s + (0.945 − 0.324i)4-s + (−0.995 + 0.0990i)5-s + (−0.277 + 0.960i)6-s + (0.518 + 0.854i)7-s + (0.879 − 0.475i)8-s + (−0.627 − 0.778i)9-s + (−0.965 + 0.261i)10-s + (0.965 + 0.261i)11-s + (−0.115 + 0.993i)12-s + (0.997 + 0.0660i)13-s + (0.652 + 0.757i)14-s + (0.340 − 0.940i)15-s + (0.789 − 0.614i)16-s + (−0.828 − 0.560i)17-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)2-s + (−0.431 + 0.901i)3-s + (0.945 − 0.324i)4-s + (−0.995 + 0.0990i)5-s + (−0.277 + 0.960i)6-s + (0.518 + 0.854i)7-s + (0.879 − 0.475i)8-s + (−0.627 − 0.778i)9-s + (−0.965 + 0.261i)10-s + (0.965 + 0.261i)11-s + (−0.115 + 0.993i)12-s + (0.997 + 0.0660i)13-s + (0.652 + 0.757i)14-s + (0.340 − 0.940i)15-s + (0.789 − 0.614i)16-s + (−0.828 − 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.251908272 + 1.390013365i\)
\(L(\frac12)\) \(\approx\) \(3.251908272 + 1.390013365i\)
\(L(1)\) \(\approx\) \(1.734479374 + 0.4029484228i\)
\(L(1)\) \(\approx\) \(1.734479374 + 0.4029484228i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.986 - 0.164i)T \)
3 \( 1 + (-0.431 + 0.901i)T \)
5 \( 1 + (-0.995 + 0.0990i)T \)
7 \( 1 + (0.518 + 0.854i)T \)
11 \( 1 + (0.965 + 0.261i)T \)
13 \( 1 + (0.997 + 0.0660i)T \)
17 \( 1 + (-0.828 - 0.560i)T \)
19 \( 1 + (-0.180 - 0.983i)T \)
23 \( 1 + (0.991 - 0.131i)T \)
29 \( 1 + (0.245 + 0.969i)T \)
31 \( 1 + (0.789 - 0.614i)T \)
37 \( 1 + (-0.846 - 0.533i)T \)
41 \( 1 + (0.401 + 0.915i)T \)
43 \( 1 + (0.0495 - 0.998i)T \)
47 \( 1 + (-0.789 + 0.614i)T \)
53 \( 1 + (0.148 + 0.988i)T \)
59 \( 1 + (-0.0825 + 0.996i)T \)
61 \( 1 + (0.701 - 0.712i)T \)
67 \( 1 + (0.148 + 0.988i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.846 + 0.533i)T \)
79 \( 1 + (0.956 - 0.293i)T \)
83 \( 1 + (-0.277 + 0.960i)T \)
89 \( 1 + (0.277 - 0.960i)T \)
97 \( 1 + (-0.0165 - 0.999i)T \)
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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

−22.87961728901065876777584679402, −22.71780141542060218261095406574, −21.279107230153526628057134721680, −20.48357222556169851357567481695, −19.56752027492207924517094448412, −19.10603417153541631828060094826, −17.684895515853362034586434285008, −16.9490311478815104840508504284, −16.23976018613936250083118414211, −15.218618419191933645742705809152, −14.27917687074941763673662341539, −13.56533181298822556062081787122, −12.75671191888563100345431850464, −11.84458207472805764944022846583, −11.25338996321969589618821208434, −10.61809495498270444093531539335, −8.44705668332689275601528975149, −7.98018754967918786480820008662, −6.8242776064855153386045881939, −6.418262707516036467763081009821, −5.10581494098972215771830899410, −4.12697151732024333833989640167, −3.39658863250482417434077853986, −1.76782335125683283342084698496, −0.862918709098195122441050093559, 0.975450525858856803680301973225, 2.616485117555162429781348180818, 3.59505255675992502033256987714, 4.48234081171173140018519304294, 5.05678405790090426435983706855, 6.27162929871178968958098611521, 7.01260398124800919195733383672, 8.54316943946004332152219107010, 9.25978081939780909991070163031, 10.79174075740026375604991240060, 11.28477317001163323191803697806, 11.832825772885299447016141049157, 12.72878209350942247250120897069, 14.03127439157152246103334331727, 14.910389461351095988953542274009, 15.47323485145457925360279710903, 15.98908563690572860203591207461, 17.026435443448228308928123797836, 18.1329383066412447080145809965, 19.273478246460944816303552137178, 20.09237544887209052691965931243, 20.83344072162853381834617151950, 21.616046484980274367067617440363, 22.41598591776830344038775881192, 22.8852667482109567660814184693

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