Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 15-s − 8·17-s + 4·19-s + 4·23-s − 4·25-s + 27-s + 5·29-s + 7·31-s − 33-s − 8·37-s + 4·41-s − 10·43-s − 45-s + 6·47-s − 8·51-s + 53-s + 55-s + 4·57-s − 9·59-s + 2·61-s − 2·67-s + 4·69-s + 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s − 1.94·17-s + 0.917·19-s + 0.834·23-s − 4/5·25-s + 0.192·27-s + 0.928·29-s + 1.25·31-s − 0.174·33-s − 1.31·37-s + 0.624·41-s − 1.52·43-s − 0.149·45-s + 0.875·47-s − 1.12·51-s + 0.137·53-s + 0.134·55-s + 0.529·57-s − 1.17·59-s + 0.256·61-s − 0.244·67-s + 0.481·69-s + 0.712·71-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: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.27677889718574165044874312562, −6.88298392907106955004455777593, −6.11774569157545328725063197884, −5.09746025032864527661845271787, −4.55539665874592638557555275084, −3.81285328418228069411832437057, −2.97322664841172772784956805325, −2.34343031954050965312769990274, −1.28148958065700333529783039154, 0, 1.28148958065700333529783039154, 2.34343031954050965312769990274, 2.97322664841172772784956805325, 3.81285328418228069411832437057, 4.55539665874592638557555275084, 5.09746025032864527661845271787, 6.11774569157545328725063197884, 6.88298392907106955004455777593, 7.27677889718574165044874312562

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