Properties

Label 2-1152-1.1-c1-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 4·11-s − 6·13-s − 6·17-s − 4·23-s − 5·25-s + 4·29-s + 10·31-s − 2·37-s + 2·41-s − 8·43-s + 12·47-s − 3·49-s − 12·53-s − 4·59-s − 2·61-s − 4·67-s + 4·71-s − 10·73-s − 8·77-s − 6·79-s + 12·83-s − 2·89-s − 12·91-s − 6·97-s + 4·101-s − 10·103-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.834·23-s − 25-s + 0.742·29-s + 1.79·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s − 1.64·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.474·71-s − 1.17·73-s − 0.911·77-s − 0.675·79-s + 1.31·83-s − 0.211·89-s − 1.25·91-s − 0.609·97-s + 0.398·101-s − 0.985·103-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: \(1\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.514926545655033706616286267341, −8.352864246059164846839031639199, −7.86636283231227873209414758068, −7.00186001179257888349709337610, −5.96427857852485389909948133454, −4.86024035915155668889978626292, −4.46736447254899856037450817253, −2.79778517840079752431830788770, −2.02071693225675944817521065685, 0, 2.02071693225675944817521065685, 2.79778517840079752431830788770, 4.46736447254899856037450817253, 4.86024035915155668889978626292, 5.96427857852485389909948133454, 7.00186001179257888349709337610, 7.86636283231227873209414758068, 8.352864246059164846839031639199, 9.514926545655033706616286267341

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