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Properties

Label 1-71-71.69-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.253 - 0.967i$
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  + (−0.995 − 0.0896i)2-s + (0.134 + 0.990i)3-s + (0.983 + 0.178i)4-s + (−0.809 + 0.587i)5-s + (−0.0448 − 0.998i)6-s + (−0.858 − 0.512i)7-s + (−0.963 − 0.266i)8-s + (−0.963 + 0.266i)9-s + (0.858 − 0.512i)10-s + (0.550 + 0.834i)11-s + (−0.0448 + 0.998i)12-s + (0.550 − 0.834i)13-s + (0.809 + 0.587i)14-s + (−0.691 − 0.722i)15-s + (0.936 + 0.351i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0896i)2-s + (0.134 + 0.990i)3-s + (0.983 + 0.178i)4-s + (−0.809 + 0.587i)5-s + (−0.0448 − 0.998i)6-s + (−0.858 − 0.512i)7-s + (−0.963 − 0.266i)8-s + (−0.963 + 0.266i)9-s + (0.858 − 0.512i)10-s + (0.550 + 0.834i)11-s + (−0.0448 + 0.998i)12-s + (0.550 − 0.834i)13-s + (0.809 + 0.587i)14-s + (−0.691 − 0.722i)15-s + (0.936 + 0.351i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.253 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08743623197 - 0.1132515440i\)
\(L(\frac12)\) \(\approx\) \(0.08743623197 - 0.1132515440i\)
\(L(1)\) \(\approx\) \(0.4572099586 + 0.1151993391i\)
\(L(1)\) \(\approx\) \(0.4572099586 + 0.1151993391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.995 - 0.0896i)T \)
3 \( 1 + (0.134 + 0.990i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.858 - 0.512i)T \)
11 \( 1 + (0.550 + 0.834i)T \)
13 \( 1 + (0.550 - 0.834i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.691 + 0.722i)T \)
23 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.936 + 0.351i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (0.753 + 0.657i)T \)
47 \( 1 + (-0.134 + 0.990i)T \)
53 \( 1 + (-0.983 + 0.178i)T \)
59 \( 1 + (0.0448 - 0.998i)T \)
61 \( 1 + (-0.858 + 0.512i)T \)
67 \( 1 + (-0.983 - 0.178i)T \)
73 \( 1 + (-0.995 - 0.0896i)T \)
79 \( 1 + (-0.963 - 0.266i)T \)
83 \( 1 + (-0.0448 + 0.998i)T \)
89 \( 1 + (0.983 - 0.178i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−31.62635031551516152807150264343, −30.51388055129005833628887498863, −29.29481023526585021813753320559, −28.53215663637552573668175071987, −27.54508125221914191911102672558, −26.131707600648236394063043513947, −25.34347724609640976689539480977, −24.13405972459684280142210256619, −23.573263238628959669781878319636, −21.66930011730544781788227164896, −20.01304934006268619940295971959, −19.317524327914043716723410839387, −18.68598324129717096482801505435, −17.17425782142487045480099257823, −16.25736320695414147571282816131, −15.049822810066007384289520898, −13.24345340717941160162113509008, −12.05781258812880093356467194432, −11.148994546238695079762486911551, −9.0192663994688678960867272054, −8.52875367822829479555778104373, −7.021929100033263831881718780298, −6.030281099528495712287457484615, −3.32913886656312944863982889910, −1.48812069011986222769608861384, 0.09858967467005083569976818003, 2.87239185002346406872530532162, 4.05180645332166074868029143176, 6.38018253580202713016793722238, 7.6711772403789784489748454800, 9.05098553776709804273328798157, 10.21656132116326256717927567282, 10.969402672805909928090769473237, 12.37728535728506993635382813827, 14.52685682676032000229956607659, 15.61863297182467137161147291689, 16.33483746739222924585794227967, 17.59633034758321987114166335438, 19.01877986302203946614973667969, 19.9984625916835825570862965260, 20.69833816061381318357751206440, 22.4181479649164401141660123118, 23.105375958656394884648890902902, 25.13733084837867050536693220788, 25.940767675320679304424903235356, 26.95196237986041929667560519925, 27.55407366302838517542811938929, 28.63946778648441612621115249553, 29.91926737724459306670640453749, 30.98839649716862492643459646029

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