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 + 9-s + 4·13-s − 4·17-s − 4·19-s + 4·23-s − 5·25-s + 27-s − 2·29-s − 8·31-s + 6·37-s + 4·39-s − 12·41-s − 4·43-s + 8·47-s − 4·51-s − 6·53-s − 4·57-s + 12·59-s + 4·61-s + 4·67-s + 4·69-s − 12·71-s − 8·73-s − 5·75-s − 16·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s − 25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s + 1.16·47-s − 0.560·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.512·61-s + 0.488·67-s + 0.481·69-s − 1.42·71-s − 0.936·73-s − 0.577·75-s − 1.80·79-s + 1/9·81-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.28686591472127527786509677170, −6.80717644372764036286297833432, −6.02156688538073099338410987442, −5.35982149610580813477694962966, −4.33864086726607298800982946552, −3.87871215971599332584395706305, −3.06428767599390184135382995660, −2.14230305969654438640210615615, −1.43166838474237182947887898341, 0, 1.43166838474237182947887898341, 2.14230305969654438640210615615, 3.06428767599390184135382995660, 3.87871215971599332584395706305, 4.33864086726607298800982946552, 5.35982149610580813477694962966, 6.02156688538073099338410987442, 6.80717644372764036286297833432, 7.28686591472127527786509677170

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