Properties

Label 2-231-1.1-c1-0-4
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  − 0.618·2-s + 3-s − 1.61·4-s + 5-s − 0.618·6-s + 7-s + 2.23·8-s + 9-s − 0.618·10-s + 11-s − 1.61·12-s + 3.47·13-s − 0.618·14-s + 15-s + 1.85·16-s + 5.23·17-s − 0.618·18-s − 6.70·19-s − 1.61·20-s + 21-s − 0.618·22-s + 5.70·23-s + 2.23·24-s − 4·25-s − 2.14·26-s + 27-s − 1.61·28-s + ⋯
L(s)  = 1  − 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.447·5-s − 0.252·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.195·10-s + 0.301·11-s − 0.467·12-s + 0.962·13-s − 0.165·14-s + 0.258·15-s + 0.463·16-s + 1.26·17-s − 0.145·18-s − 1.53·19-s − 0.361·20-s + 0.218·21-s − 0.131·22-s + 1.19·23-s + 0.456·24-s − 0.800·25-s − 0.420·26-s + 0.192·27-s − 0.305·28-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

\(L(1)\) \(\approx\) \(1.166632767\)
\(L(\frac12)\) \(\approx\) \(1.166632767\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
5 \( 1 - T + 5T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 - 0.236T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 4.52T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 6.76T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 - 9.70T + 97T^{2} \)
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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

−12.41423514810206826750858491646, −10.92398971985737103802843587805, −10.13712429767744275515780916615, −9.089557933844557850256785970912, −8.514660235077902213508523967115, −7.51605687488813425627016717498, −6.04744333367486623437160774081, −4.72873507257686043114874995132, −3.50835522546679541734792285367, −1.53195910092535386464608359172, 1.53195910092535386464608359172, 3.50835522546679541734792285367, 4.72873507257686043114874995132, 6.04744333367486623437160774081, 7.51605687488813425627016717498, 8.514660235077902213508523967115, 9.089557933844557850256785970912, 10.13712429767744275515780916615, 10.92398971985737103802843587805, 12.41423514810206826750858491646

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