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 + 2·5-s + 9-s − 2·13-s − 2·15-s − 6·17-s + 4·19-s + 4·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 10·37-s + 2·39-s + 10·41-s + 12·43-s + 2·45-s − 8·47-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s − 10·61-s − 4·65-s + 12·67-s − 4·69-s − 4·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s + 0.298·45-s − 1.16·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.481·69-s − 0.474·71-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.37858363838248446160078701342, −6.57789868247435997998295651334, −5.91049326221092939323680020702, −5.45341968887645466689109288381, −4.65714616628027431432206560326, −4.00965934400221477305823637249, −2.85571758992652827608417970940, −2.15150344614261598632778619468, −1.25423182140621920036242037665, 0, 1.25423182140621920036242037665, 2.15150344614261598632778619468, 2.85571758992652827608417970940, 4.00965934400221477305823637249, 4.65714616628027431432206560326, 5.45341968887645466689109288381, 5.91049326221092939323680020702, 6.57789868247435997998295651334, 7.37858363838248446160078701342

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