Properties

Label 1-231-231.170-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: $\chi_{231} (170, \cdot )$
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

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

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