Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 2·11-s + 3·13-s + 2·15-s + 8·17-s + 19-s + 8·23-s − 25-s + 27-s − 4·29-s + 3·31-s − 2·33-s + 37-s + 3·39-s + 6·41-s − 11·43-s + 2·45-s + 6·47-s + 8·51-s + 12·53-s − 4·55-s + 57-s − 4·59-s + 6·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.832·13-s + 0.516·15-s + 1.94·17-s + 0.229·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.538·31-s − 0.348·33-s + 0.164·37-s + 0.480·39-s + 0.937·41-s − 1.67·43-s + 0.298·45-s + 0.875·47-s + 1.12·51-s + 1.64·53-s − 0.539·55-s + 0.132·57-s − 0.520·59-s + 0.768·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.757504713\)
\(L(\frac12)\) \(\approx\) \(3.757504713\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.59070171086489945413384046837, −7.25084496613277172040708881818, −6.20565953329243398205547425304, −5.63634745551612070262727888164, −5.12850841208142859912625736773, −4.10496358485525709791193006329, −3.22841734310084284583453266603, −2.76170314076069094288108859683, −1.68431114231995019739344442312, −0.977588273663314206280701756714, 0.977588273663314206280701756714, 1.68431114231995019739344442312, 2.76170314076069094288108859683, 3.22841734310084284583453266603, 4.10496358485525709791193006329, 5.12850841208142859912625736773, 5.63634745551612070262727888164, 6.20565953329243398205547425304, 7.25084496613277172040708881818, 7.59070171086489945413384046837

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