↑

Properties

Label 1-71-71.60-r0-0-0
Degree $1$
Conductor $71$
Sign $0.985 - 0.167i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.550 − 0.834i)2-s + (−0.995 − 0.0896i)3-s + (−0.393 + 0.919i)4-s + (−0.809 + 0.587i)5-s + (0.473 + 0.880i)6-s + (0.936 − 0.351i)7-s + (0.983 − 0.178i)8-s + (0.983 + 0.178i)9-s + (0.936 + 0.351i)10-s + (0.134 + 0.990i)11-s + (0.473 − 0.880i)12-s + (0.134 − 0.990i)13-s + (−0.809 − 0.587i)14-s + (0.858 − 0.512i)15-s + (−0.691 − 0.722i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.550 − 0.834i)2-s + (−0.995 − 0.0896i)3-s + (−0.393 + 0.919i)4-s + (−0.809 + 0.587i)5-s + (0.473 + 0.880i)6-s + (0.936 − 0.351i)7-s + (0.983 − 0.178i)8-s + (0.983 + 0.178i)9-s + (0.936 + 0.351i)10-s + (0.134 + 0.990i)11-s + (0.473 − 0.880i)12-s + (0.134 − 0.990i)13-s + (−0.809 − 0.587i)14-s + (0.858 − 0.512i)15-s + (−0.691 − 0.722i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5022418497 - 0.04225044022i\)
\(L(\frac12)\) \(\approx\) \(0.5022418497 - 0.04225044022i\)
\(L(1)\) \(\approx\) \(0.5850023349 - 0.1045315979i\)
\(L(1)\) \(\approx\) \(0.5850023349 - 0.1045315979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.550 - 0.834i)T \)
3 \( 1 + (-0.995 - 0.0896i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.936 - 0.351i)T \)
11 \( 1 + (0.134 + 0.990i)T \)
13 \( 1 + (0.134 - 0.990i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.858 + 0.512i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (-0.691 + 0.722i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.0448 + 0.998i)T \)
47 \( 1 + (-0.995 + 0.0896i)T \)
53 \( 1 + (-0.393 - 0.919i)T \)
59 \( 1 + (0.473 - 0.880i)T \)
61 \( 1 + (0.936 + 0.351i)T \)
67 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.550 - 0.834i)T \)
79 \( 1 + (0.983 - 0.178i)T \)
83 \( 1 + (0.473 - 0.880i)T \)
89 \( 1 + (-0.393 - 0.919i)T \)
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

−31.99474597507779259012219373129, −30.81824138213936072787354133351, −29.04743367705035692621010959732, −28.29038678531542141440344565071, −27.22905568233329316959042062435, −26.79600184477534366418567359084, −24.78451869197396342388459382503, −24.11602615511977390100545815069, −23.38082847541274304226689147419, −22.067321836200283778338619811948, −20.65562667881465090783118541620, −19.00128449605477507469754032833, −18.23367304765826640820985071679, −16.86147881496415067423175908227, −16.25747764265063588423248910891, −15.17087658905677671805955734829, −13.69095164971718572347277579572, −11.795573202734633478916445854118, −11.12703959474375946261242900932, −9.347960881049088889585188566998, −8.17316806467308181138774871645, −6.883951532775545266791072728330, −5.39957806896613595858610887723, −4.48926765923084703630696303179, −0.99964047100731491852855988345, 1.42419748500900618169178887854, 3.61379953220026192072395050243, 4.9805112532336300992831070214, 7.15843771179582372507101362387, 8.032296464944753965728955072773, 10.09105432151358881663215679142, 10.92119975654603115946925796280, 11.850033612817295388714251670754, 12.84103455803585727389507096013, 14.73567503969682152492091135494, 16.186505130782688874128967457262, 17.61896155994954684216065419570, 18.02781402807106596028678336642, 19.39610225001109266336805200727, 20.499247484275963949210061238065, 21.76057814895221825290054018756, 22.84320326391392451155164763090, 23.576412507117560548499759997577, 25.27929498413543645248794286876, 26.84339700282502818629066252466, 27.49004093612898979668509890829, 28.23511604480047875321549222896, 29.5520409813526017849142575574, 30.46280066688169321699078377267, 31.02265706409826897506267134494

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