↑

Properties

Label 1-71-71.63-r1-0-0
Degree $1$
Conductor $71$
Sign $0.0297 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.393 − 0.919i)3-s + (0.858 + 0.512i)4-s + (0.309 + 0.951i)5-s + (0.134 + 0.990i)6-s + (0.0448 − 0.998i)7-s + (−0.691 − 0.722i)8-s + (−0.691 + 0.722i)9-s + (−0.0448 − 0.998i)10-s + (−0.983 + 0.178i)11-s + (0.134 − 0.990i)12-s + (−0.983 − 0.178i)13-s + (−0.309 + 0.951i)14-s + (0.753 − 0.657i)15-s + (0.473 + 0.880i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.393 − 0.919i)3-s + (0.858 + 0.512i)4-s + (0.309 + 0.951i)5-s + (0.134 + 0.990i)6-s + (0.0448 − 0.998i)7-s + (−0.691 − 0.722i)8-s + (−0.691 + 0.722i)9-s + (−0.0448 − 0.998i)10-s + (−0.983 + 0.178i)11-s + (0.134 − 0.990i)12-s + (−0.983 − 0.178i)13-s + (−0.309 + 0.951i)14-s + (0.753 − 0.657i)15-s + (0.473 + 0.880i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0297 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0297 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.0297 + 0.999i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.0297 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2426264199 + 0.2499569139i\)
\(L(\frac12)\) \(\approx\) \(0.2426264199 + 0.2499569139i\)
\(L(1)\) \(\approx\) \(0.5058265180 - 0.05972650241i\)
\(L(1)\) \(\approx\) \(0.5058265180 - 0.05972650241i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.963 - 0.266i)T \)
3 \( 1 + (-0.393 - 0.919i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.0448 - 0.998i)T \)
11 \( 1 + (-0.983 + 0.178i)T \)
13 \( 1 + (-0.983 - 0.178i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.753 + 0.657i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (-0.473 + 0.880i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.550 + 0.834i)T \)
47 \( 1 + (0.393 - 0.919i)T \)
53 \( 1 + (-0.858 + 0.512i)T \)
59 \( 1 + (-0.134 + 0.990i)T \)
61 \( 1 + (0.0448 + 0.998i)T \)
67 \( 1 + (-0.858 - 0.512i)T \)
73 \( 1 + (-0.963 - 0.266i)T \)
79 \( 1 + (-0.691 - 0.722i)T \)
83 \( 1 + (0.134 - 0.990i)T \)
89 \( 1 + (0.858 - 0.512i)T \)
97 \( 1 + (0.900 + 0.433i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.63829180529169368575645550439, −29.459257642008564472233141590813, −28.52959806526908401967502544330, −28.05913005310905198359706440341, −26.91264377903480667188122131478, −25.89752292243540634876089556620, −24.72885260686647999384873293419, −23.81866226438341308251741137866, −22.08939657142088545250152996535, −21.0038766301683685459667531359, −20.20798679618809404951334593202, −18.629300589664833661135024261361, −17.55877192031750504625530187856, −16.4355518884038620458317197718, −15.79143805940653917077893235348, −14.57228838079409132947285232056, −12.42308379375625248408574561907, −11.32619219119491897666908926702, −9.80085411385194653143633551992, −9.206563679038621373360455513856, −7.86822208401419909557290518194, −5.800803363315235559708537448097, −5.04603377487500421143821226836, −2.51526105615436962294602864728, −0.24122653773325117853817158712, 1.61449575157304473122858136064, 3.09134527357856677316655009427, 5.83413070519927345826115808387, 7.36406155532242172602249041575, 7.69119575884552530984710375155, 9.95557662396741283940384157150, 10.70393031969370548254577821843, 11.964741877307611888906599820683, 13.263780910975975918007599367303, 14.65318431793824665230476328431, 16.432820981494618221501147279446, 17.48245786919492611420519173359, 18.24613007518567477159552480681, 19.22082800253780036520263179001, 20.26238671429453640578184590977, 21.724188419596345977889647737781, 23.04831228510025136759162288462, 24.13066177108557830831137714988, 25.45274896667722867501328230918, 26.262772896620229301767670891993, 27.30329989642534141673433368653, 28.71274119981739278012702506081, 29.599400453176283118253700901496, 30.07826908646760919228800569444, 31.23022353261666281836795175526

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