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Properties

Label 1-571-571.236-r0-0-0
Degree $1$
Conductor $571$
Sign $0.735 + 0.678i$
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  + (−0.926 − 0.376i)2-s + (0.993 + 0.110i)3-s + (0.716 + 0.697i)4-s + (−0.821 + 0.569i)5-s + (−0.879 − 0.475i)6-s + (0.945 + 0.324i)7-s + (−0.401 − 0.915i)8-s + (0.975 + 0.218i)9-s + (0.975 − 0.218i)10-s + (−0.298 − 0.954i)11-s + (0.635 + 0.771i)12-s + (0.451 + 0.892i)13-s + (−0.754 − 0.656i)14-s + (−0.879 + 0.475i)15-s + (0.0275 + 0.999i)16-s + (0.0275 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (0.993 + 0.110i)3-s + (0.716 + 0.697i)4-s + (−0.821 + 0.569i)5-s + (−0.879 − 0.475i)6-s + (0.945 + 0.324i)7-s + (−0.401 − 0.915i)8-s + (0.975 + 0.218i)9-s + (0.975 − 0.218i)10-s + (−0.298 − 0.954i)11-s + (0.635 + 0.771i)12-s + (0.451 + 0.892i)13-s + (−0.754 − 0.656i)14-s + (−0.879 + 0.475i)15-s + (0.0275 + 0.999i)16-s + (0.0275 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.735 + 0.678i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.735 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175510413 + 0.4593909964i\)
\(L(\frac12)\) \(\approx\) \(1.175510413 + 0.4593909964i\)
\(L(1)\) \(\approx\) \(0.9886833467 + 0.1205344107i\)
\(L(1)\) \(\approx\) \(0.9886833467 + 0.1205344107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.926 - 0.376i)T \)
3 \( 1 + (0.993 + 0.110i)T \)
5 \( 1 + (-0.821 + 0.569i)T \)
7 \( 1 + (0.945 + 0.324i)T \)
11 \( 1 + (-0.298 - 0.954i)T \)
13 \( 1 + (0.451 + 0.892i)T \)
17 \( 1 + (0.0275 + 0.999i)T \)
19 \( 1 + (0.993 + 0.110i)T \)
23 \( 1 + (-0.401 + 0.915i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.451 - 0.892i)T \)
41 \( 1 + (-0.821 + 0.569i)T \)
43 \( 1 + (0.975 - 0.218i)T \)
47 \( 1 + (0.851 - 0.523i)T \)
53 \( 1 + (-0.926 + 0.376i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (0.137 + 0.990i)T \)
67 \( 1 + (0.137 - 0.990i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.998 + 0.0550i)T \)
79 \( 1 + (-0.962 - 0.272i)T \)
83 \( 1 + (0.851 - 0.523i)T \)
89 \( 1 + (0.0275 + 0.999i)T \)
97 \( 1 + (0.716 + 0.697i)T \)
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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

−23.62306577485592691745007956351, −22.48045252005886543556399057563, −20.66136246730754933517479344870, −20.462857741464284561972952029608, −20.113944188981443923643307928714, −18.84337014582342992129389792432, −18.22903433240545165379024008871, −17.46904288931960658886861819658, −16.238696785133072825963296368352, −15.64419342578265330543166075091, −14.87885465521232820708422966506, −14.141101711363795186882123240658, −12.9476151923665452123212161609, −11.9457024424816897550031955826, −10.97438603605396616493124675340, −9.97935541882616238853611032812, −9.07462487790092412579793752898, −8.25754100776943967808856011333, −7.59211025288487233652713864368, −7.12767941918694313867407160922, −5.35262759097875369886872709238, −4.48079172193181319811036982525, −3.147315119614671499833951610092, −1.91132096379518356555366351986, −0.875814196969459645808513521824, 1.39325043643237300352871311767, 2.34744869514604193793201065445, 3.49034678721719632186413986127, 4.04731866560220145483500107859, 5.857003060273993072613675594101, 7.302644524147562764201885062021, 7.80054385789284559868060099158, 8.59508449148534728466387635241, 9.28778828302926251080300655057, 10.51126244717860589031339452421, 11.23542404102795283829608034085, 11.88449777205526215262412869332, 13.148546829100111840000590821876, 14.194561282050176677732115374346, 15.011812616125946134593115592652, 15.81508524730167076841481120873, 16.50774619379369709291010972586, 17.83651307367937122809359404603, 18.709398890840437871767107167915, 18.97360371560868880499842367802, 19.94822544132219485838668570795, 20.65238028734006923759850406952, 21.54978056588211672878819283236, 21.99025982763486854649923415173, 23.84429313924409663669546416877

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