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Properties

Label 1-71-71.50-r0-0-0
Degree $1$
Conductor $71$
Sign $0.854 + 0.519i$
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  + (−0.393 + 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 − 0.722i)4-s + (0.309 − 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (0.936 − 0.351i)8-s + (0.936 + 0.351i)9-s + (0.753 + 0.657i)10-s + (−0.963 + 0.266i)11-s + (−0.550 − 0.834i)12-s + (−0.963 − 0.266i)13-s + (0.309 + 0.951i)14-s + (0.473 − 0.880i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.393 + 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 − 0.722i)4-s + (0.309 − 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (0.936 − 0.351i)8-s + (0.936 + 0.351i)9-s + (0.753 + 0.657i)10-s + (−0.963 + 0.266i)11-s + (−0.550 − 0.834i)12-s + (−0.963 − 0.266i)13-s + (0.309 + 0.951i)14-s + (0.473 − 0.880i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.854 + 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9927453351 + 0.2783634664i\)
\(L(\frac12)\) \(\approx\) \(0.9927453351 + 0.2783634664i\)
\(L(1)\) \(\approx\) \(1.066175206 + 0.2762380642i\)
\(L(1)\) \(\approx\) \(1.066175206 + 0.2762380642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.393 + 0.919i)T \)
3 \( 1 + (0.983 + 0.178i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.753 - 0.657i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (-0.963 - 0.266i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.473 + 0.880i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (-0.0448 - 0.998i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.995 - 0.0896i)T \)
47 \( 1 + (0.983 - 0.178i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (-0.550 - 0.834i)T \)
61 \( 1 + (0.753 + 0.657i)T \)
67 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (-0.393 + 0.919i)T \)
79 \( 1 + (0.936 - 0.351i)T \)
83 \( 1 + (-0.550 - 0.834i)T \)
89 \( 1 + (-0.691 + 0.722i)T \)
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

−31.126078240607613544093499685, −30.67880477212795118796422101817, −29.54559992000993890816595490694, −28.55176362915648332676796969011, −26.84864647272310643241579637761, −26.6115190549968675876749706508, −25.334650036803291857735925977017, −24.11811243637060771197914591823, −22.29422908633571533296557748510, −21.47785622935370785037454371875, −20.523109862993696214718679608136, −19.28310518139032290527779650533, −18.41418919715919329983822563549, −17.68960823755824659286422319066, −15.55503096251957239545891963328, −14.31639283983521187938730870197, −13.403578444191167274331090113116, −11.96274799591227401760271039781, −10.6830594824132624638797425953, −9.51355386464829816963675254730, −8.34027184706800161099769419037, −7.169119829775527861309235338786, −4.75436770074589340832488575315, −2.84300216236873777990348007172, −2.22393705951275504771069381705, 1.73690650769953652870517947618, 4.28346597972935073180901778619, 5.33393889035050047291984954365, 7.469759417185134334006374857165, 8.155295595201770133737080008589, 9.41186386867819854535254350736, 10.411831036329084101119863521181, 12.85524735074896637353611279393, 13.81754634364142539534421181004, 14.88677716102963617366936726276, 15.97104870536622840184699563144, 17.14907904688218069005362530692, 18.162831103870807539816931863379, 19.70666740947048815690914235846, 20.48085527893486629591152239503, 21.74304228073683392752523748746, 23.58557984178042227797379602342, 24.369501467713579195629568793, 25.20158029305844605950874043670, 26.35274422161759878373489990299, 27.13213789515530215541384010184, 28.19504883424701036658844760794, 29.57866871878770169460618984742, 31.308854795476399747177124061807, 31.76819159998185472439448524712

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