Properties

Label 1-231-231.41-r1-0-0
Degree $1$
Conductor $231$
Sign $-0.642 + 0.766i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)8-s + 10-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)20-s − 23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)26-s + (−0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + 32-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)8-s + 10-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)20-s − 23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)26-s + (−0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05079022407 - 0.1088802401i\)
\(L(\frac12)\) \(\approx\) \(-0.05079022407 - 0.1088802401i\)
\(L(1)\) \(\approx\) \(0.8131617523 - 0.3577713247i\)
\(L(1)\) \(\approx\) \(0.8131617523 - 0.3577713247i\)

Euler product

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

−26.18854981968552772520288038008, −25.90350269573063200599835177850, −24.62913602605969164114938939309, −24.03815031795512198295137631566, −23.35432438003286377218272798731, −21.971004050649324815649301952132, −21.432552895564066818349211867441, −20.31676094391087840887992728169, −19.08778206983859899953097684900, −17.94042486845205442679934339784, −17.015846246445554064976649133899, −16.404856734470747970400174227531, −15.394491185343873633069250917378, −14.36540539319409759610468933376, −13.325686139691844281727924876346, −12.70479478922396321996787720655, −11.49487257733767601394241550995, −9.83930083804480262121635198012, −8.84786025623232392266493400200, −8.14327789494402461722517881430, −6.73110222143338036317669227275, −5.844821259582254593450211842306, −4.688510039406127260637236034850, −3.83295807935404963970252570604, −1.82535528059364087363231551022, 0.03323362594111421016734511667, 1.82022793635962962744746510755, 2.91254937253073720193119182382, 3.91902293130488345520276539822, 5.37270136201944514228376824892, 6.350479744691892461151553225758, 7.82304469950405146221295755504, 9.17289074986761798682445093864, 10.24735629759433495263712817363, 10.86870890122811900468635479392, 11.93594105075611760828006177508, 13.01983874234713188263703232199, 13.95604738945096381654480143376, 14.73724032312059561972014071564, 15.79601729158426576400772376784, 17.467314720426858098307494134027, 18.182132747715453418148447715298, 18.97037488263012459986564017739, 19.994924627971205129575465104361, 20.89595199156664866412994491851, 21.807997778916995703501127432133, 22.678810305734830693312120786472, 23.18379863808353335711504024943, 24.52330181096724821865970911942, 25.618455784642679287224744724098

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