Properties

Label 1-231-231.86-r1-0-0
Degree $1$
Conductor $231$
Sign $0.952 - 0.304i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9420642588 - 0.1470772544i\)
\(L(\frac12)\) \(\approx\) \(0.9420642588 - 0.1470772544i\)
\(L(1)\) \(\approx\) \(0.7130413205 - 0.3489784172i\)
\(L(1)\) \(\approx\) \(0.7130413205 - 0.3489784172i\)

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

−26.15664182220708424105735294797, −25.211227818734967209254035448629, −24.187448030269542019491951790988, −23.5304239248145708384322949363, −22.60960817352638573621558988654, −21.91038104735060478811308199511, −20.5880159128974676305569731087, −19.22824031703696258923537536620, −18.70402166507103788271687561390, −17.50225765734099382846755060096, −16.54603265239551600136816416569, −15.72466943480061981829563883459, −14.77847861005267928022973445708, −14.12416982622342160017634899693, −12.77908773026472907623968139339, −11.87632171266950661286504820857, −10.58166271731847384209999054772, −9.27858851772314461330816620973, −8.29234942872777675886405872634, −7.21670892238458415064990933365, −6.58622712823859193118404911795, −5.00964986151435111785088720262, −4.18150374610984625124961667447, −2.81281861204364662791827613796, −0.41939472037458911080833027066, 0.88718739351370152408407801263, 2.42642181806869209989756794244, 3.74081520423759104141157540810, 4.57899478183915449214368872170, 5.82102108622469074372292408737, 7.64161510013236060290418824013, 8.48069307950400372847340303478, 9.638772271294915754942552481184, 10.69717611751223313309324355046, 11.63939838199639717795504135361, 12.54208564206859879416667513280, 13.21917334913191234283173688101, 14.679417759055496179997924274057, 15.36392509437347613997981525069, 16.88153015989200667799701945823, 17.616002797051694508046793236128, 19.13866830744428622799845171959, 19.37019631126171040713756768149, 20.43930790428955083498591159287, 21.28093282820545051870693900859, 22.28276327953499334058802718270, 23.23368514972484475102349554460, 23.86976190279890474970265621835, 25.05320595119501961876727752783, 26.453140758852687204292665454578

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