Properties

Label 1-231-231.194-r1-0-0
Degree $1$
Conductor $231$
Sign $0.734 + 0.678i$
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  + (0.669 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174770576 + 0.4598407734i\)
\(L(\frac12)\) \(\approx\) \(1.174770576 + 0.4598407734i\)
\(L(1)\) \(\approx\) \(1.059824302 - 0.2773196546i\)
\(L(1)\) \(\approx\) \(1.059824302 - 0.2773196546i\)

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

−26.02579295690867745295007029907, −24.74000149949030794348595193352, −24.22862155353394287643589011462, −23.16335997949134854491160603807, −22.627195374982833605556699019761, −21.53218605121623895805257253950, −20.467353788012250666802221780046, −19.619986391362310108118239623583, −18.31472209487354651128198294353, −17.271707441001900737318523210966, −16.36723060493812660768386908119, −15.28445504338092244843360716361, −15.00906255076349168759445626035, −13.439668342932344339034389106529, −12.78771829454674495107139395879, −11.722551394396589655463973072184, −10.74401609094589629046297694649, −8.92605325261371381529601842780, −8.14188437502445702985210279569, −7.174838663920358248920162124505, −6.06161513606296128350026498289, −4.79782836007281418564388931124, −3.92435787373010276894329867855, −2.722952878996339733351068784346, −0.34840046972137953153425949092, 1.31997720104922816572255819115, 2.83138445145136414284850988261, 3.91194193564392128659698213942, 4.78361003037088080527830017965, 6.21155959039288133130073250945, 7.31742180102221474782273026935, 8.75112314970598966971803057747, 9.85213926715956713352658589329, 11.16748712944976309302181795223, 11.60382069896253202139015957903, 12.67902082686731704646057210971, 13.746521491929765441166635363167, 14.685208198854810503514581014101, 15.58595862343116372907856531805, 16.52668273998037643736655820718, 18.18818571074889685520800122761, 18.94051870020023833197732480072, 19.75159481248469025641826608818, 20.635323193759298218568796779045, 21.557585462380445864785494922928, 22.59628038198550363162650771679, 23.28209498928514886977532061615, 24.036248690766260292154628436955, 25.0477216006165107875112026233, 26.49891832503322764642981289451

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