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Properties

Label 1-59-59.44-r1-0-0
Degree $1$
Conductor $59$
Sign $0.204 - 0.978i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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  + (−0.976 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.204 - 0.978i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3922754316 - 0.3187308658i\)
\(L(\frac12)\) \(\approx\) \(0.3922754316 - 0.3187308658i\)
\(L(1)\) \(\approx\) \(0.5575307804 + 0.01663318153i\)
\(L(1)\) \(\approx\) \(0.5575307804 + 0.01663318153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.976 + 0.214i)T \)
3 \( 1 + (-0.161 + 0.986i)T \)
5 \( 1 + (0.267 - 0.963i)T \)
7 \( 1 + (-0.725 + 0.687i)T \)
11 \( 1 + (-0.647 - 0.762i)T \)
13 \( 1 + (0.947 - 0.319i)T \)
17 \( 1 + (-0.725 - 0.687i)T \)
19 \( 1 + (-0.370 - 0.928i)T \)
23 \( 1 + (0.994 - 0.108i)T \)
29 \( 1 + (0.976 + 0.214i)T \)
31 \( 1 + (0.370 - 0.928i)T \)
37 \( 1 + (-0.796 - 0.605i)T \)
41 \( 1 + (-0.994 - 0.108i)T \)
43 \( 1 + (-0.647 + 0.762i)T \)
47 \( 1 + (-0.267 - 0.963i)T \)
53 \( 1 + (0.0541 + 0.998i)T \)
61 \( 1 + (-0.976 + 0.214i)T \)
67 \( 1 + (-0.796 + 0.605i)T \)
71 \( 1 + (0.267 + 0.963i)T \)
73 \( 1 + (0.561 - 0.827i)T \)
79 \( 1 + (-0.161 - 0.986i)T \)
83 \( 1 + (-0.468 - 0.883i)T \)
89 \( 1 + (-0.976 - 0.214i)T \)
97 \( 1 + (0.561 + 0.827i)T \)
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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

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

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