Properties

Label 2-231-1.1-c1-0-10
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  + 2.47·2-s + 3-s + 4.11·4-s − 2.58·5-s + 2.47·6-s − 7-s + 5.22·8-s + 9-s − 6.39·10-s − 11-s + 4.11·12-s − 5.87·13-s − 2.47·14-s − 2.58·15-s + 4.70·16-s + 7.51·17-s + 2.47·18-s − 2.35·19-s − 10.6·20-s − 21-s − 2.47·22-s + 6.94·23-s + 5.22·24-s + 1.69·25-s − 14.5·26-s + 27-s − 4.11·28-s + ⋯
L(s)  = 1  + 1.74·2-s + 0.577·3-s + 2.05·4-s − 1.15·5-s + 1.00·6-s − 0.377·7-s + 1.84·8-s + 0.333·9-s − 2.02·10-s − 0.301·11-s + 1.18·12-s − 1.62·13-s − 0.660·14-s − 0.668·15-s + 1.17·16-s + 1.82·17-s + 0.582·18-s − 0.540·19-s − 2.38·20-s − 0.218·21-s − 0.527·22-s + 1.44·23-s + 1.06·24-s + 0.339·25-s − 2.84·26-s + 0.192·27-s − 0.777·28-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

\(L(1)\) \(\approx\) \(2.944315718\)
\(L(\frac12)\) \(\approx\) \(2.944315718\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.47T + 2T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
13 \( 1 + 5.87T + 13T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 + 2.35T + 19T^{2} \)
23 \( 1 - 6.94T + 23T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + 3.66T + 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 - 7.40T + 43T^{2} \)
47 \( 1 - 7.53T + 47T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 + 0.926T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 4.45T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−12.52404471879143102964196798798, −11.69988601541872017996300816489, −10.64145954225033776958527833627, −9.331081393855998552863132008413, −7.60328759004254524416029840779, −7.30779760444084841078467261484, −5.69653518869871741849245878104, −4.61930078937300391216756519466, −3.59901998740036571373751238840, −2.67958341900950104318063369812, 2.67958341900950104318063369812, 3.59901998740036571373751238840, 4.61930078937300391216756519466, 5.69653518869871741849245878104, 7.30779760444084841078467261484, 7.60328759004254524416029840779, 9.331081393855998552863132008413, 10.64145954225033776958527833627, 11.69988601541872017996300816489, 12.52404471879143102964196798798

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