Properties

Label 2-1152-1.1-c1-0-15
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

−9.289315418586462146642251685690, −8.671123783015345129139988405634, −7.76841700352222764456930738236, −6.81542041238937589948552667187, −6.04519620219891673101591163967, −5.19382499916710112913058943849, −3.97633443717640489814889299763, −3.13420541549434036877759112013, −1.86962407001750988737926530376, 0, 1.86962407001750988737926530376, 3.13420541549434036877759112013, 3.97633443717640489814889299763, 5.19382499916710112913058943849, 6.04519620219891673101591163967, 6.81542041238937589948552667187, 7.76841700352222764456930738236, 8.671123783015345129139988405634, 9.289315418586462146642251685690

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