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Properties

Label 1-231-231.26-r0-0-0
Degree $1$
Conductor $231$
Sign $0.865 + 0.501i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(1.07275\)
Root analytic conductor: \(1.07275\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (0:\ ),\ 0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6691878766 + 0.1799336646i\)
\(L(\frac12)\) \(\approx\) \(0.6691878766 + 0.1799336646i\)
\(L(1)\) \(\approx\) \(0.6621802317 + 0.1712510466i\)
\(L(1)\) \(\approx\) \(0.6621802317 + 0.1712510466i\)

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

−26.76684163859932320373271191593, −25.480310246153910372242054025456, −24.532791556294491885145432434880, −23.29907252472967498422006268737, −22.53952447047055903105428188622, −21.302340275838194800772732687400, −20.5888034780302310781798041543, −19.60516122523223424254351611227, −18.90290821434503579480666145092, −18.10230877845229835262872254781, −16.58904905043722013860107400541, −16.382461566074969767685944708940, −14.89758619833580961040168485540, −13.68641992367641826537258774941, −12.29683793021557418299944077736, −11.90666302601924764965850213544, −10.808491393668652400035427609794, −9.72699161189963249612462068645, −8.67463224892122773715508540322, −7.78501507712635837257431218282, −6.79079899443138639259604904164, −4.80861757083378364572188979789, −3.814281981379816017461478590, −2.619533176712288310219765675658, −1.00502483562950367018993573948, 0.894343515385008130759771219229, 2.88561523861597460157245259838, 4.4207286805632558901851588783, 5.594782826607669650925778705338, 6.854867485170385964089510179409, 7.771046030841072345685304217505, 8.50849773375657654186010681735, 9.81677100000614074092779046675, 10.75991611123025247462082007385, 11.78455569771192568355973886558, 13.107345869448588735835143326501, 14.39790252340902815729399306676, 15.33324424469698712904113416262, 15.81163586637330369589826413607, 17.09149583048284301299676525782, 17.78455599516002357715705649647, 19.04806994645621117301708698728, 19.50383642700236263711704780526, 20.509337681425936833432049429864, 22.0112912141750288868413486777, 23.083162412817644799310174376959, 23.65978454169069952830571572795, 24.63351716077968639118814310343, 25.56865662951675646873447151797, 26.45578665837385639874979341569

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