Properties

Degree 1
Conductor 71
Sign $0.985 + 0.167i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.985 + 0.167i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (58, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.985 + 0.167i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5022418497 + 0.04225044022i$
$L(\frac12,\chi)$  $\approx$  $0.5022418497 + 0.04225044022i$
$L(\chi,1)$  $\approx$  0.5850023349 + 0.1045315979i
$L(1,\chi)$  $\approx$  0.5850023349 + 0.1045315979i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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