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 + 3·11-s + 4·13-s + 15-s + 4·19-s − 8·23-s − 4·25-s − 27-s + 3·29-s − 5·31-s − 3·33-s − 8·37-s − 4·39-s − 8·41-s + 6·43-s − 45-s + 10·47-s − 9·53-s − 3·55-s − 4·57-s + 5·59-s − 10·61-s − 4·65-s + 6·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.258·15-s + 0.917·19-s − 1.66·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.522·33-s − 1.31·37-s − 0.640·39-s − 1.24·41-s + 0.914·43-s − 0.149·45-s + 1.45·47-s − 1.23·53-s − 0.404·55-s − 0.529·57-s + 0.650·59-s − 1.28·61-s − 0.496·65-s + 0.733·67-s + 0.963·69-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−7.37545831882754496725994633134, −6.59244702976879714109273997668, −5.98544091675367820335028549234, −5.46860108618752601540212875297, −4.46259714765689536582002995470, −3.84247787894769981059793340920, −3.32544313718735246652437895821, −1.94839091245589974945757712100, −1.19800120186953262104149046971, 0, 1.19800120186953262104149046971, 1.94839091245589974945757712100, 3.32544313718735246652437895821, 3.84247787894769981059793340920, 4.46259714765689536582002995470, 5.46860108618752601540212875297, 5.98544091675367820335028549234, 6.59244702976879714109273997668, 7.37545831882754496725994633134

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