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Properties

Label 1-231-231.128-r0-0-0
Degree $1$
Conductor $231$
Sign $-0.110 - 0.993i$
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  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.309 − 0.951i)13-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.913 − 0.406i)26-s + (−0.809 + 0.587i)29-s + (−0.978 − 0.207i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (−0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.309 − 0.951i)13-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.913 − 0.406i)26-s + (−0.809 + 0.587i)29-s + (−0.978 − 0.207i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ -0.110 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240679698 - 1.386476114i\)
\(L(\frac12)\) \(\approx\) \(1.240679698 - 1.386476114i\)
\(L(1)\) \(\approx\) \(1.347949294 - 0.8498443814i\)
\(L(1)\) \(\approx\) \(1.347949294 - 0.8498443814i\)

Euler product

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

−26.41850668265302839205660210376, −25.375728913380576041034976246864, −24.88542155690948262384648798769, −23.85108063166763496907856317528, −22.8590976157139211013077995168, −22.06142199522793226404520485397, −21.19462844896241537773790244593, −20.504808459827914063110956226099, −18.76845433632275825870182128237, −18.04366841409730850087844986723, −16.748960544654047642156377654981, −16.477617131901837995678623284997, −14.869796975816974025777506539541, −14.3137935400715314082872017737, −13.41415131765983395970060212219, −12.4426755545876669234395763335, −11.35238803732416967734348884811, −9.88359906886309342932759579131, −8.96021582283089553193192301809, −7.6163575200365609633813295076, −6.62388215982485614679165754917, −5.69526376866671191141668839495, −4.686158922256562551811754382791, −3.307948901432682373895222878743, −2.02450197437646375074369272883, 1.23868040365338136789284342442, 2.4710229363253680357677744386, 3.61583072428689012711584401023, 5.15515442288509216724740568066, 5.69363225357411919159031415917, 7.06226453341098659243650893528, 8.77088822650154853330598080479, 9.80208915488483258038169126039, 10.55836397898319887434340607504, 11.686103609560081562020203861500, 12.95081637879736491919589218509, 13.29548226000699477981075105014, 14.564325616274919084988356794121, 15.26955839706646406819111598124, 16.75836972586267802735123135385, 17.765285774661494347760823404074, 18.67423685291984241734798970821, 19.86885595362442619859953949336, 20.51972432846639178886032508003, 21.676276151142150992728645255098, 21.99551828057075406100371250049, 23.23512782797490899184867167589, 24.09754224250381738731752486646, 25.05864425822405285918644277778, 25.904771341033856607906767155

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