Properties

Label 2-231-1.1-c1-0-3
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  − 0.167·2-s − 3-s − 1.97·4-s + 3.80·5-s + 0.167·6-s − 7-s + 0.665·8-s + 9-s − 0.637·10-s + 11-s + 1.97·12-s + 3.80·13-s + 0.167·14-s − 3.80·15-s + 3.83·16-s + 0.334·17-s − 0.167·18-s + 8.13·19-s − 7.50·20-s + 21-s − 0.167·22-s − 1.66·23-s − 0.665·24-s + 9.47·25-s − 0.637·26-s − 27-s + 1.97·28-s + ⋯
L(s)  = 1  − 0.118·2-s − 0.577·3-s − 0.985·4-s + 1.70·5-s + 0.0683·6-s − 0.377·7-s + 0.235·8-s + 0.333·9-s − 0.201·10-s + 0.301·11-s + 0.569·12-s + 1.05·13-s + 0.0447·14-s − 0.982·15-s + 0.958·16-s + 0.0812·17-s − 0.0394·18-s + 1.86·19-s − 1.67·20-s + 0.218·21-s − 0.0357·22-s − 0.347·23-s − 0.135·24-s + 1.89·25-s − 0.124·26-s − 0.192·27-s + 0.372·28-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

\(L(1)\) \(\approx\) \(1.065687792\)
\(L(\frac12)\) \(\approx\) \(1.065687792\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.167T + 2T^{2} \)
5 \( 1 - 3.80T + 5T^{2} \)
13 \( 1 - 3.80T + 13T^{2} \)
17 \( 1 - 0.334T + 17T^{2} \)
19 \( 1 - 8.13T + 19T^{2} \)
23 \( 1 + 1.66T + 23T^{2} \)
29 \( 1 - 0.195T + 29T^{2} \)
31 \( 1 + 9.94T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 6.27T + 41T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 7.94T + 53T^{2} \)
59 \( 1 - 3.74T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 0.139T + 67T^{2} \)
71 \( 1 - 4.66T + 71T^{2} \)
73 \( 1 - 4.19T + 73T^{2} \)
79 \( 1 - 3.33T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 - 0.0560T + 97T^{2} \)
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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

−12.39136965295135208323663305317, −11.07930172290141520520214586072, −9.950552289827991469279434977737, −9.555886725570303644764308080199, −8.572362247806168425889216766756, −6.95392807657252279307714040797, −5.76481845648844578550047148204, −5.24952291857016122984277096353, −3.55798727359313927513609986045, −1.41280128727805385968207146263, 1.41280128727805385968207146263, 3.55798727359313927513609986045, 5.24952291857016122984277096353, 5.76481845648844578550047148204, 6.95392807657252279307714040797, 8.572362247806168425889216766756, 9.555886725570303644764308080199, 9.950552289827991469279434977737, 11.07930172290141520520214586072, 12.39136965295135208323663305317

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