Properties

Label 2-231-1.1-c1-0-0
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  − 2.36·2-s − 3-s + 3.57·4-s − 3.93·5-s + 2.36·6-s − 7-s − 3.72·8-s + 9-s + 9.29·10-s + 11-s − 3.57·12-s − 3.93·13-s + 2.36·14-s + 3.93·15-s + 1.63·16-s + 4.72·17-s − 2.36·18-s + 4.78·19-s − 14.0·20-s + 21-s − 2.36·22-s + 2.72·23-s + 3.72·24-s + 10.5·25-s + 9.29·26-s − 27-s − 3.57·28-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.577·3-s + 1.78·4-s − 1.76·5-s + 0.964·6-s − 0.377·7-s − 1.31·8-s + 0.333·9-s + 2.94·10-s + 0.301·11-s − 1.03·12-s − 1.09·13-s + 0.631·14-s + 1.01·15-s + 0.409·16-s + 1.14·17-s − 0.556·18-s + 1.09·19-s − 3.14·20-s + 0.218·21-s − 0.503·22-s + 0.567·23-s + 0.759·24-s + 2.10·25-s + 1.82·26-s − 0.192·27-s − 0.675·28-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

\(L(1)\) \(\approx\) \(0.2884527795\)
\(L(\frac12)\) \(\approx\) \(0.2884527795\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.36T + 2T^{2} \)
5 \( 1 + 3.93T + 5T^{2} \)
13 \( 1 + 3.93T + 13T^{2} \)
17 \( 1 - 4.72T + 17T^{2} \)
19 \( 1 - 4.78T + 19T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 - 1.15T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 - 0.430T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + 8.78T + 47T^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 3.21T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 5.44T + 79T^{2} \)
83 \( 1 + 2.84T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−11.95242471111440881028458137742, −11.13471626413881062728809971276, −10.18777369309842366265042442160, −9.330020223226769115731956494068, −8.120950972724659484615139109448, −7.49340801158776844309309128524, −6.70907230934053884435300035857, −4.85450489928686870441731589615, −3.21821698187565017901290515740, −0.76038619061696059228109838303, 0.76038619061696059228109838303, 3.21821698187565017901290515740, 4.85450489928686870441731589615, 6.70907230934053884435300035857, 7.49340801158776844309309128524, 8.120950972724659484615139109448, 9.330020223226769115731956494068, 10.18777369309842366265042442160, 11.13471626413881062728809971276, 11.95242471111440881028458137742

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