Properties

Label 1-231-231.230-r1-0-0
Degree $1$
Conductor $231$
Sign $1$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s + 13-s + 16-s − 17-s + 19-s + 20-s − 23-s + 25-s + 26-s + 29-s − 31-s + 32-s − 34-s + 37-s + 38-s + 40-s − 41-s − 43-s − 46-s + 47-s + 50-s + 52-s − 53-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s + 13-s + 16-s − 17-s + 19-s + 20-s − 23-s + 25-s + 26-s + 29-s − 31-s + 32-s − 34-s + 37-s + 38-s + 40-s − 41-s − 43-s − 46-s + 47-s + 50-s + 52-s − 53-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{231} (230, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.723357944\)
\(L(\frac12)\) \(\approx\) \(4.723357944\)
\(L(1)\) \(\approx\) \(2.480419453\)
\(L(1)\) \(\approx\) \(2.480419453\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−25.74052528617984478999023986334, −25.13739018047372777018482742890, −24.18490570707199724084538417312, −23.33384233284743557293196360861, −22.13847229515960441696236400515, −21.764893805090190547490063279859, −20.57417757980053627684394307461, −20.04972809235110408714692247372, −18.50988493016836789549312085731, −17.61355916607801357414439824135, −16.40104108465268915018985597654, −15.63523893556330825609929974819, −14.40648126080685617079911553763, −13.64441623671850535329869063546, −12.97623250922333672985441146463, −11.73918873370749471311740903432, −10.77006358058981667811237925317, −9.7306520279211782798732374269, −8.37033606754077383515621819097, −6.89815843648292663662783449546, −6.05121256024719143898565116891, −5.11569946981700576973382910279, −3.841352560581366627200601996248, −2.56455585443005839963354558759, −1.41283231189260459546431496489, 1.41283231189260459546431496489, 2.56455585443005839963354558759, 3.841352560581366627200601996248, 5.11569946981700576973382910279, 6.05121256024719143898565116891, 6.89815843648292663662783449546, 8.37033606754077383515621819097, 9.7306520279211782798732374269, 10.77006358058981667811237925317, 11.73918873370749471311740903432, 12.97623250922333672985441146463, 13.64441623671850535329869063546, 14.40648126080685617079911553763, 15.63523893556330825609929974819, 16.40104108465268915018985597654, 17.61355916607801357414439824135, 18.50988493016836789549312085731, 20.04972809235110408714692247372, 20.57417757980053627684394307461, 21.764893805090190547490063279859, 22.13847229515960441696236400515, 23.33384233284743557293196360861, 24.18490570707199724084538417312, 25.13739018047372777018482742890, 25.74052528617984478999023986334

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