Properties

Label 1-231-231.53-r1-0-0
Degree $1$
Conductor $231$
Sign $0.0869 - 0.996i$
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  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.809 + 0.587i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.978 + 0.207i)26-s + (−0.309 − 0.951i)29-s + (−0.104 + 0.994i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.809 + 0.587i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.978 + 0.207i)26-s + (−0.309 − 0.951i)29-s + (−0.104 + 0.994i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0869 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0869 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.0869 - 0.996i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.0869 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3220830940 - 0.2951828628i\)
\(L(\frac12)\) \(\approx\) \(0.3220830940 - 0.2951828628i\)
\(L(1)\) \(\approx\) \(0.5938365724 + 0.1083384421i\)
\(L(1)\) \(\approx\) \(0.5938365724 + 0.1083384421i\)

Euler product

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

−26.38784831519150638509972501186, −25.594303638907810660171866657195, −24.41287827302341374899683588051, −24.05451095127898807564778320117, −22.27466472726206433271332404942, −21.48347936148576226571429793522, −20.516798851241476614711247527747, −19.78284105329379904790185814337, −18.98727529685461355493097822255, −17.659624075254734317037793877553, −17.14438347078003293164921471554, −16.152148046987367744269683057481, −15.28122049224549481391563496936, −13.64474184893901046764788947960, −12.675789302479814168027347257653, −11.79533826363515120288688736958, −10.81485898145313313354083752646, −9.436911887481508203810647869398, −9.04038574651764899024490350614, −7.799816950510886808435802593233, −6.79529400273611884274914251584, −5.23930409672523923165382264313, −3.96083437823842922034124934625, −2.40153148127621099047194428989, −1.22590189618200263057802036324, 0.19960962580671951909943482260, 2.008992430897901770855667795948, 3.11791584111105591624108779264, 5.044259095681924673293839235188, 6.289297271546192513051170910747, 7.148932021836784461416204881742, 8.0612467804752794826328142822, 9.35684220105135727641542835233, 10.25319363731647162956061893605, 11.04560632082371158929278570508, 12.15620345582993108875650333366, 13.77071177106562279184263281940, 14.72544886971802117511900282147, 15.44358003212809786222521799702, 16.528022287396710381076276859243, 17.62067473193999337790671030323, 18.22789554413205056697000199184, 19.17444409529250791972699002814, 20.000830027155229009255347713832, 21.08374346438566490459358991990, 22.42088323554692001446560374839, 23.03480404750903245548902744150, 24.460284572457607724045013141459, 24.96447627069582794938972767945, 26.09194936690289687906948251337

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