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Properties

Label 1-71-71.66-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.227 + 0.973i$
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  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.227 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6161970962 + 0.7771435573i\)
\(L(\frac12)\) \(\approx\) \(0.6161970962 + 0.7771435573i\)
\(L(1)\) \(\approx\) \(0.6379722702 + 0.3860381238i\)
\(L(1)\) \(\approx\) \(0.6379722702 + 0.3860381238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 - T \)
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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

−30.517291206498589894522020686344, −30.04435469671140086458372968260, −28.626065513472572176040205833534, −28.034544154078307554749208251263, −27.41161388388693861029654149466, −25.33530695500850736281861200860, −24.89724538910108381852561369965, −23.525768133318335277311588157094, −22.061294822517917258595622086810, −21.034422069598277736961259325041, −19.96598948216892405588582822906, −18.62019123295511553684879564353, −17.66280569321187848471407215037, −17.009389197375947957835654011188, −15.73062616468545755668605483654, −13.5565327580796263796392818148, −12.27154798552667845788245219260, −11.72264814583399590000273201712, −10.27566131155529075525687506356, −8.82804286663655742734793723595, −7.78313430160187352998521237658, −6.09326922588331183036223874011, −4.548897170076375684934512441381, −2.02207616605795179406374537635, −0.86621291619646336015057874243, 1.34978251308141194259994876501, 4.03200143176244117370396716253, 5.85867644747621312078664591887, 6.659120097717553970665267578835, 8.25432598435135501218682764741, 9.83103542550343403605641088051, 10.76940851249892844451491128656, 11.5574212519695105801697557574, 14.11035977065245850139920574749, 14.83451086681726850791858228319, 16.29320057469442299123483917678, 17.138159595153891836797419633391, 18.08119070652669940183562832164, 19.103908621613143223528543752963, 20.7592277904177245554892248376, 21.867128499946278458132574750755, 23.234734756576254825363272463692, 23.923049149939138509181289964543, 25.49744945462730553900491232206, 26.51062761044869304580315918879, 27.3117365736268544706732539935, 28.13423790036134074650482036798, 29.48836990307329111631242841005, 30.22834253759509168005550044288, 32.26798303280532990074805809405

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