Properties

Label 2-1152-1.1-c1-0-3
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 + 4·7-s − 2·11-s + 2·13-s + 2·17-s − 2·19-s + 4·23-s − 25-s + 6·29-s − 8·35-s + 10·37-s + 6·41-s − 6·43-s − 8·47-s + 9·49-s + 6·53-s + 4·55-s + 14·59-s + 2·61-s − 4·65-s − 10·67-s + 12·71-s + 14·73-s − 8·77-s + 8·79-s − 6·83-s − 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.35·35-s + 1.64·37-s + 0.937·41-s − 0.914·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 1.82·59-s + 0.256·61-s − 0.496·65-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.911·77-s + 0.900·79-s − 0.658·83-s − 0.433·85-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

\(L(1)\) \(\approx\) \(1.647878569\)
\(L(\frac12)\) \(\approx\) \(1.647878569\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−9.878602275413527189048173292252, −8.675368536957604275874860747441, −8.088659191412945349881968287052, −7.63242500478736670765641851926, −6.52256885432452015704757780960, −5.34050418818343429420051907291, −4.62995407509459210959268189869, −3.73824894896581631226090928700, −2.44235807183272353930685105042, −1.02724387350145696940295704376, 1.02724387350145696940295704376, 2.44235807183272353930685105042, 3.73824894896581631226090928700, 4.62995407509459210959268189869, 5.34050418818343429420051907291, 6.52256885432452015704757780960, 7.63242500478736670765641851926, 8.088659191412945349881968287052, 8.675368536957604275874860747441, 9.878602275413527189048173292252

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