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Properties

Label 1-71-71.65-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.954 - 0.298i$
Analytic cond. $7.63000$
Root an. cond. $7.63000$
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  + (−0.0448 − 0.998i)2-s + (0.753 − 0.657i)3-s + (−0.995 + 0.0896i)4-s + (0.309 − 0.951i)5-s + (−0.691 − 0.722i)6-s + (0.963 − 0.266i)7-s + (0.134 + 0.990i)8-s + (0.134 − 0.990i)9-s + (−0.963 − 0.266i)10-s + (−0.473 − 0.880i)11-s + (−0.691 + 0.722i)12-s + (−0.473 + 0.880i)13-s + (−0.309 − 0.951i)14-s + (−0.393 − 0.919i)15-s + (0.983 − 0.178i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (0.753 − 0.657i)3-s + (−0.995 + 0.0896i)4-s + (0.309 − 0.951i)5-s + (−0.691 − 0.722i)6-s + (0.963 − 0.266i)7-s + (0.134 + 0.990i)8-s + (0.134 − 0.990i)9-s + (−0.963 − 0.266i)10-s + (−0.473 − 0.880i)11-s + (−0.691 + 0.722i)12-s + (−0.473 + 0.880i)13-s + (−0.309 − 0.951i)14-s + (−0.393 − 0.919i)15-s + (0.983 − 0.178i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.954 - 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2930633360 - 1.918886635i\)
\(L(\frac12)\) \(\approx\) \(0.2930633360 - 1.918886635i\)
\(L(1)\) \(\approx\) \(0.8133839635 - 1.075329608i\)
\(L(1)\) \(\approx\) \(0.8133839635 - 1.075329608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.963 - 0.266i)T \)
11 \( 1 + (-0.473 - 0.880i)T \)
13 \( 1 + (-0.473 + 0.880i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.393 + 0.919i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.983 - 0.178i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.936 - 0.351i)T \)
47 \( 1 + (-0.753 - 0.657i)T \)
53 \( 1 + (0.995 + 0.0896i)T \)
59 \( 1 + (0.691 - 0.722i)T \)
61 \( 1 + (0.963 + 0.266i)T \)
67 \( 1 + (0.995 - 0.0896i)T \)
73 \( 1 + (-0.0448 - 0.998i)T \)
79 \( 1 + (0.134 + 0.990i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.995 - 0.0896i)T \)
97 \( 1 + (0.900 + 0.433i)T \)
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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

−32.16734167801798615292391144050, −30.89731827839326467827759179783, −30.34154925300009918407014563203, −28.11333896183191799349109051913, −27.3070542425949158459617848183, −26.25852158116488382353264171569, −25.56764921557401369320658090059, −24.582105752768743648085993957087, −23.16031755743113803862908919882, −22.03868499364354419502515965525, −21.15044286941131616100025705694, −19.588818594600531937223576712738, −18.224071185023162130159743705819, −17.43153089870023350951085862276, −15.793610836322004994095868569583, −14.758769135007349957417084564898, −14.45249202560701200196022109031, −12.89808994813280454931385605895, −10.65706960409053337254741445351, −9.72946901541022197398177667090, −8.23990895992328163428894269435, −7.3696258978479892431425863136, −5.57556858201303660670356045570, −4.32199483028196175570462966863, −2.5135438189422067469957523261, 1.00233994934471471410881856242, 2.15932552660799845448556180216, 3.915742847465783142163721174500, 5.41176818575117762376065384243, 7.81096197147171301246271721480, 8.677461555134311859847467638830, 9.85647192998918527409467259424, 11.56103092736952923291192515609, 12.53484237558828594242013694114, 13.74843224946580229699152047881, 14.36504145049881645210432345405, 16.59047047000506357390824476284, 17.84590617787889653863793634219, 18.84158396844461263688152008031, 19.93634271451261067603202185961, 20.93021311936118063279839606449, 21.46899666014048475068693476534, 23.56864212634451667402848621780, 24.145658567803063581543892071955, 25.5053590747463417222102010781, 26.84199285623380897183019958217, 27.71634866259145650425732268612, 29.26866454499666097271255911392, 29.571312247682724714709230705948, 31.075889403950755318774728847002

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