Properties

Label 2-1152-1.1-c1-0-8
Degree $2$
Conductor $1152$
Sign $1$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 4·11-s + 2·13-s + 4·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 6·31-s + 4·35-s − 2·37-s + 12·41-s + 12·43-s − 8·47-s − 3·49-s + 6·53-s + 8·55-s + 8·59-s − 10·61-s + 4·65-s − 8·67-s + 2·73-s + 8·77-s − 14·79-s + 12·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.07·31-s + 0.676·35-s − 0.328·37-s + 1.87·41-s + 1.82·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.07·55-s + 1.04·59-s − 1.28·61-s + 0.496·65-s − 0.977·67-s + 0.234·73-s + 0.911·77-s − 1.57·79-s + 1.31·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1152} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.256534198\)
\(L(\frac12)\) \(\approx\) \(2.256534198\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.782683048374999514747740510300, −9.017830298727617035965492530881, −8.239699972575940949478402632684, −7.36183700646228938196129724807, −6.11969843230623172416001421015, −5.87618764591883614635582577159, −4.54054370996283917506505788819, −3.72962610687736412807879856876, −2.22088860133337487727877949530, −1.31006766181093226223763545704, 1.31006766181093226223763545704, 2.22088860133337487727877949530, 3.72962610687736412807879856876, 4.54054370996283917506505788819, 5.87618764591883614635582577159, 6.11969843230623172416001421015, 7.36183700646228938196129724807, 8.239699972575940949478402632684, 9.017830298727617035965492530881, 9.782683048374999514747740510300

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