Properties

Degree 1
Conductor 59
Sign $0.204 + 0.978i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.976 − 0.214i)2-s + (−0.161 − 0.986i)3-s + (0.907 + 0.419i)4-s + (0.267 + 0.963i)5-s + (−0.0541 + 0.998i)6-s + (−0.725 − 0.687i)7-s + (−0.796 − 0.605i)8-s + (−0.947 + 0.319i)9-s + (−0.0541 − 0.998i)10-s + (−0.647 + 0.762i)11-s + (0.267 − 0.963i)12-s + (0.947 + 0.319i)13-s + (0.561 + 0.827i)14-s + (0.907 − 0.419i)15-s + (0.647 + 0.762i)16-s + (−0.725 + 0.687i)17-s + ⋯
L(s,χ)  = 1  + (−0.976 − 0.214i)2-s + (−0.161 − 0.986i)3-s + (0.907 + 0.419i)4-s + (0.267 + 0.963i)5-s + (−0.0541 + 0.998i)6-s + (−0.725 − 0.687i)7-s + (−0.796 − 0.605i)8-s + (−0.947 + 0.319i)9-s + (−0.0541 − 0.998i)10-s + (−0.647 + 0.762i)11-s + (0.267 − 0.963i)12-s + (0.947 + 0.319i)13-s + (0.561 + 0.827i)14-s + (0.907 − 0.419i)15-s + (0.647 + 0.762i)16-s + (−0.725 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.204 + 0.978i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.204 + 0.978i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $0.204 + 0.978i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (55, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ 0.204 + 0.978i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3922754316 + 0.3187308658i$
$L(\frac12,\chi)$  $\approx$  $0.3922754316 + 0.3187308658i$
$L(\chi,1)$  $\approx$  0.5575307804 + 0.01663318153i
$L(1,\chi)$  $\approx$  0.5575307804 + 0.01663318153i

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

−32.51427342713824547289210982859, −31.47741384170499032570033197604, −29.32269964431601760267585735625, −28.5479126037797783375849085249, −27.88028886030918019145818309259, −26.65426567602175388873965280798, −25.67094969925410559866373111093, −24.68506816722660280794132559063, −23.27121437123920125550735071467, −21.61978023621150630061943062823, −20.74782201256590307435694026458, −19.65320290398637785872364820095, −18.22921055257980894938635737703, −16.96837507628703096833038325641, −15.960033821217645236304055326097, −15.44303026735765426567858071189, −13.34156618137171396294173217852, −11.62554349904119447864768884899, −10.41979899813552615282658939187, −9.09178465163137334064944441236, −8.581108865702181644511538335902, −6.26373614219449847621976219442, −5.13196871029766745560913557866, −2.87006523062819967719048716970, −0.37114540523326124990295206665, 1.67840896938782529700022983168, 3.17981210935411602293863435487, 6.372731824906534790582801899532, 7.00751226612438082067405444707, 8.372675333841308542199533918304, 10.143326206567309661911046732696, 10.99952214229798471330279212626, 12.513510209143564137536246478401, 13.69075218897576484452513529308, 15.40970447027859063030650573750, 16.9339759816578998964697916381, 17.90849821680228129153523038068, 18.81714448573105021415142753978, 19.67191936043155249564589284767, 21.026108216509723338216559955143, 22.71550074063658934600594683041, 23.64205488510397619317557765243, 25.366320234425734932758030092531, 25.825856485651666618958197198026, 26.94111450271537109556458777665, 28.65949385035627055038369898787, 29.135420622572624794366616504179, 30.323286152629580591403765084849, 30.92856410828828292367892227673, 33.19243935135992827908983850122

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