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Properties

Label 1-71-71.40-r0-0-0
Degree $1$
Conductor $71$
Sign $0.992 - 0.118i$
Analytic cond. $0.329722$
Root an. cond. $0.329722$
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  + (0.936 − 0.351i)2-s + (0.858 + 0.512i)3-s + (0.753 − 0.657i)4-s + (−0.809 + 0.587i)5-s + (0.983 + 0.178i)6-s + (−0.550 − 0.834i)7-s + (0.473 − 0.880i)8-s + (0.473 + 0.880i)9-s + (−0.550 + 0.834i)10-s + (−0.691 + 0.722i)11-s + (0.983 − 0.178i)12-s + (−0.691 − 0.722i)13-s + (−0.809 − 0.587i)14-s + (−0.995 + 0.0896i)15-s + (0.134 − 0.990i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (0.858 + 0.512i)3-s + (0.753 − 0.657i)4-s + (−0.809 + 0.587i)5-s + (0.983 + 0.178i)6-s + (−0.550 − 0.834i)7-s + (0.473 − 0.880i)8-s + (0.473 + 0.880i)9-s + (−0.550 + 0.834i)10-s + (−0.691 + 0.722i)11-s + (0.983 − 0.178i)12-s + (−0.691 − 0.722i)13-s + (−0.809 − 0.587i)14-s + (−0.995 + 0.0896i)15-s + (0.134 − 0.990i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.992 - 0.118i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.992 - 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.648510356 - 0.09777596411i\)
\(L(\frac12)\) \(\approx\) \(1.648510356 - 0.09777596411i\)
\(L(1)\) \(\approx\) \(1.712728870 - 0.09310965841i\)
\(L(1)\) \(\approx\) \(1.712728870 - 0.09310965841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.936 - 0.351i)T \)
3 \( 1 + (0.858 + 0.512i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.550 - 0.834i)T \)
11 \( 1 + (-0.691 + 0.722i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.995 - 0.0896i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (0.134 + 0.990i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.963 - 0.266i)T \)
47 \( 1 + (0.858 - 0.512i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (0.983 - 0.178i)T \)
61 \( 1 + (-0.550 + 0.834i)T \)
67 \( 1 + (0.753 - 0.657i)T \)
73 \( 1 + (0.936 - 0.351i)T \)
79 \( 1 + (0.473 - 0.880i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (0.753 + 0.657i)T \)
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

−31.72005812511298241684283993582, −31.202925332494410669037063188392, −29.744591466092853228480902901414, −28.88033173783698689341265656253, −27.118989214387573408195047000401, −25.91885575276254137467888529297, −24.95728245204885169724033517706, −24.08503657396288801250497906578, −23.2700451372622906611580627903, −21.72637349242175027684115394516, −20.77685539393446377798639333618, −19.579974176165163167892093523420, −18.69845288649671299220514563461, −16.66785738029829644401758849836, −15.65041798354645848275018666619, −14.74931917092987303822279145870, −13.37953179112894080189930180934, −12.54197240104520535081208973803, −11.549095461948454179654680899070, −9.14862172730208847528770384770, −8.03112371207408016476785220587, −6.89327839635338550943805437981, −5.26854294397934299022831162118, −3.67551042100373045775030852639, −2.45292903548172940519524044890, 2.55280809878436866489642947616, 3.66448595761880572870771595363, 4.71473923103069978711150585804, 6.82430597937355781691729221379, 7.94692462150839807651627691078, 10.15553659659031996704072763807, 10.61498597676676888480211524170, 12.46178060898309815164699603323, 13.44337246588533635596309737413, 14.924438348118775722994351594541, 15.21933751220095150002164503800, 16.66271115161748796155449951301, 18.922974423585427053390879194234, 19.76838300092031644081363353605, 20.51623460632305919033259952682, 21.78620346539179724013315766658, 22.87341997344554532439002554063, 23.658652961053090006991877059082, 25.18796238952029467449880739535, 26.197876336523399515201388190184, 27.24757076422856380831988464111, 28.53460777501220063085150345423, 30.09966638275489983746361902908, 30.5597688679744801881599880448, 31.77870460709192456011509619838

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