Properties

Label 2-231-1.1-c1-0-9
Degree $2$
Conductor $231$
Sign $1$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.52·2-s − 3-s + 4.39·4-s + 0.133·5-s − 2.52·6-s − 7-s + 6.05·8-s + 9-s + 0.337·10-s + 11-s − 4.39·12-s + 0.133·13-s − 2.52·14-s − 0.133·15-s + 6.52·16-s − 5.05·17-s + 2.52·18-s − 0.924·19-s + 0.586·20-s + 21-s + 2.52·22-s − 7.05·23-s − 6.05·24-s − 4.98·25-s + 0.337·26-s − 27-s − 4.39·28-s + ⋯
L(s)  = 1  + 1.78·2-s − 0.577·3-s + 2.19·4-s + 0.0596·5-s − 1.03·6-s − 0.377·7-s + 2.14·8-s + 0.333·9-s + 0.106·10-s + 0.301·11-s − 1.26·12-s + 0.0370·13-s − 0.675·14-s − 0.0344·15-s + 1.63·16-s − 1.22·17-s + 0.596·18-s − 0.212·19-s + 0.131·20-s + 0.218·21-s + 0.539·22-s − 1.47·23-s − 1.23·24-s − 0.996·25-s + 0.0662·26-s − 0.192·27-s − 0.830·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.635423434\)
\(L(\frac12)\) \(\approx\) \(2.635423434\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
good2 \( 1 - 2.52T + 2T^{2} \)
5 \( 1 - 0.133T + 5T^{2} \)
13 \( 1 - 0.133T + 13T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 0.924T + 19T^{2} \)
23 \( 1 + 7.05T + 23T^{2} \)
29 \( 1 - 3.86T + 29T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 - 9.98T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 3.05T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 + 4.79T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 8.92T + 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 - 7.86T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 1.20T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.31939094116399208259155120933, −11.58005232048979331673763077630, −10.77921814662342343046734326351, −9.558770604813578568531728217260, −7.81193716649814418758385755456, −6.43847728305086804383472755460, −6.08176024111541615510085979932, −4.71114374854496168174718887749, −3.89767349904112176684886328824, −2.32788722497410658387544301182, 2.32788722497410658387544301182, 3.89767349904112176684886328824, 4.71114374854496168174718887749, 6.08176024111541615510085979932, 6.43847728305086804383472755460, 7.81193716649814418758385755456, 9.558770604813578568531728217260, 10.77921814662342343046734326351, 11.58005232048979331673763077630, 12.31939094116399208259155120933

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