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Properties

Label 1-231-231.116-r0-0-0
Degree $1$
Conductor $231$
Sign $0.637 - 0.770i$
Analytic cond. $1.07275$
Root an. cond. $1.07275$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8094948737 - 0.3806730073i\)
\(L(\frac12)\) \(\approx\) \(0.8094948737 - 0.3806730073i\)
\(L(1)\) \(\approx\) \(0.7944637824 - 0.3092746289i\)
\(L(1)\) \(\approx\) \(0.7944637824 - 0.3092746289i\)

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

−26.54344455225651076370225321092, −25.43770077898168634112137367376, −24.60382849508336041053227837637, −23.70063612514742385834577213866, −23.10784025835469473787721413983, −22.135796760709125866595695312401, −20.86086270683372205964702680075, −19.768273643718651873055786463137, −18.79722612558191753021355287816, −17.97417389847733319256303776642, −16.781863212811659000017004733812, −15.97055979656579515071470807728, −15.42173652780759008944719354139, −14.14129458882731525783624937355, −13.34012007914034983191076450523, −12.10055181322529707312819885695, −11.07156130382109818488846738995, −9.50065142637253587936518882348, −8.74559149249816974846238388419, −7.65326670444130027400886672499, −6.870332246458026538963420577997, −5.480171409263038587719593085696, −4.50194930990587579368333288158, −3.3879021793614595192305735356, −0.99006661753984567568654486504, 1.047009922652531315365800007841, 2.76974582423702736201164886120, 3.67962627269968153159245383779, 4.74044698054563948521463916601, 6.28440871651602526651271089671, 7.87576091267179828637440510495, 8.51086443211071629565125545339, 9.93914777084510617242618880234, 10.823741754295152155336241284578, 11.64056924350126645902365393333, 12.577269226913103436714988536791, 13.57911166051174801550305023328, 14.71728736421148370096289040756, 15.70110215643381667663539029765, 16.95465456913321056197039260902, 18.114262849370359727051298536302, 18.80918045611406032779460139163, 19.72425333703839499919907055062, 20.47103073439953841527685723697, 21.46571243264767264672522730638, 22.585950917720843528361675387725, 23.070652620086622789178823951044, 24.10803290510051016856738390898, 25.5013966954932003977065857385, 26.667854882014913835660199812168

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