Properties

Label 1-231-231.68-r1-0-0
Degree $1$
Conductor $231$
Sign $-0.986 - 0.164i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.809 − 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.809 − 0.587i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.669 − 0.743i)26-s + (0.309 + 0.951i)29-s + (−0.913 + 0.406i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.809 − 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.809 − 0.587i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.669 − 0.743i)26-s + (0.309 + 0.951i)29-s + (−0.913 + 0.406i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ -0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08691667760 + 1.048443041i\)
\(L(\frac12)\) \(\approx\) \(-0.08691667760 + 1.048443041i\)
\(L(1)\) \(\approx\) \(0.7114554836 + 0.5763408247i\)
\(L(1)\) \(\approx\) \(0.7114554836 + 0.5763408247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−25.73954496208531855501496589224, −24.74154873883035264201621011688, −23.66014333648062307379745285237, −22.51492459954599124477521250476, −21.72627390562630586474870899145, −20.946188556901346826425070421160, −20.18095052825577174631868004684, −19.08067263327226240752098254246, −18.254132592284923309997367037523, −17.20647278344928943874360309479, −16.59819591003481495805879995890, −14.82350877958249065776457966800, −13.890989570972693391882449338993, −13.033120820655526493808513441702, −12.15251950573169604323393804105, −11.07826201481309260965550986853, −9.9361079328723560848636257657, −9.302139918654275922847099438599, −8.25019106622788068381497249531, −6.67699535387250419630064494295, −5.21523775995922097343621108922, −4.394169779371788292359447981217, −2.75099466312012905307652771501, −1.8454738955755725906606566444, −0.347249048481671270279415615431, 1.59847763692419296621304958107, 3.29358000174467762701344538673, 4.83973225132697432978836690139, 5.81157476086979037443157899242, 6.71611889394128124166817594669, 7.788919411245076042369859754731, 8.94806991731036679634982622603, 9.931034953527864311548896250458, 10.7709051244638474670130449669, 12.64250320448196931318701175860, 13.30286758173497240354497765613, 14.63582651094872716789359485831, 14.88832438155688596561916850735, 16.32956212108416992994737519952, 17.25047135050921232518133017240, 17.82730207919817254134120912373, 18.86477578169101289582298548267, 19.84718023145606709467289941336, 21.50891764276463092787813070313, 21.89955305996382014245472867868, 23.08037460277107162444911486600, 23.85840792153123241591652314511, 25.077773767136100607461047172228, 25.43710689680617638622804818813, 26.43370334391496348756071619878

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