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Properties

Label 1-71-71.7-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.999 + 0.00177i$
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  + (0.858 + 0.512i)2-s + (−0.691 + 0.722i)3-s + (0.473 + 0.880i)4-s + (−0.809 + 0.587i)5-s + (−0.963 + 0.266i)6-s + (0.995 + 0.0896i)7-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (−0.995 + 0.0896i)10-s + (−0.936 + 0.351i)11-s + (−0.963 − 0.266i)12-s + (−0.936 − 0.351i)13-s + (0.809 + 0.587i)14-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.858 + 0.512i)2-s + (−0.691 + 0.722i)3-s + (0.473 + 0.880i)4-s + (−0.809 + 0.587i)5-s + (−0.963 + 0.266i)6-s + (0.995 + 0.0896i)7-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (−0.995 + 0.0896i)10-s + (−0.936 + 0.351i)11-s + (−0.963 − 0.266i)12-s + (−0.936 − 0.351i)13-s + (0.809 + 0.587i)14-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.999 + 0.00177i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.999 + 0.00177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001297973288 + 1.463713094i\)
\(L(\frac12)\) \(\approx\) \(0.001297973288 + 1.463713094i\)
\(L(1)\) \(\approx\) \(0.7957045419 + 0.8634934866i\)
\(L(1)\) \(\approx\) \(0.7957045419 + 0.8634934866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.858 + 0.512i)T \)
3 \( 1 + (-0.691 + 0.722i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.995 + 0.0896i)T \)
11 \( 1 + (-0.936 + 0.351i)T \)
13 \( 1 + (-0.936 - 0.351i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.134 + 0.990i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (0.550 + 0.834i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (0.691 + 0.722i)T \)
53 \( 1 + (-0.473 + 0.880i)T \)
59 \( 1 + (0.963 + 0.266i)T \)
61 \( 1 + (0.995 - 0.0896i)T \)
67 \( 1 + (-0.473 - 0.880i)T \)
73 \( 1 + (0.858 + 0.512i)T \)
79 \( 1 + (-0.0448 + 0.998i)T \)
83 \( 1 + (-0.963 - 0.266i)T \)
89 \( 1 + (0.473 - 0.880i)T \)
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

−30.862309257730118723753977045060, −29.99926694836157913548542995771, −28.69844510743280535129827393408, −28.17598316863915660985857089802, −26.83798225579101758582560182099, −24.64953614092332023139092443615, −24.0258420165539805014732974405, −23.46912024735067128248195581065, −22.12810108609909051069807069328, −21.03254312948549589669208275861, −19.78399620168998019927063675027, −18.87745718380299933751641676304, −17.46260428423915021933912463745, −16.104974662084349574006257438479, −14.83682804563374442599083116711, −13.41801406789758470760639744121, −12.430334260534708805540461955692, −11.51282018344417786479176432890, −10.600275705545732710649822081, −8.28161532185448016585419164730, −6.98104999287732286634744498801, −5.29115255687814846053872947220, −4.4808625866327722393671741019, −2.270886921230317601226048435459, −0.59913463222098977603554087031, 2.96843453407237996636391651073, 4.51791269508967166046511580658, 5.32206484663810631871987130734, 7.02624794977909596484590909867, 8.11436346552599974766278545110, 10.301753480417092433075576338041, 11.511918433663419206487672339769, 12.236552678315350148423136089495, 14.13121461944100132756712561116, 15.21814632917579844629157011429, 15.791047373340366911772734864814, 17.20864361645630257330572707583, 18.21211167936059497377188038080, 20.24801705798925440621994411637, 21.21959933610633944809448998059, 22.271246012311076992497482909218, 23.18828563524809305500907438331, 23.904982555367418268163675134883, 25.297711371189861261991506301238, 26.9280604428416406748804928342, 27.11239964360698114573575023478, 28.85264183321392945077832534088, 29.997886376751214071051770993609, 31.242240722851148627115941711700, 31.807174147970593640105262088901

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