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 − 4·5-s + 9-s − 4·11-s − 4·13-s + 4·15-s + 4·19-s + 11·25-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 6·37-s + 4·39-s + 4·43-s − 4·45-s + 8·47-s + 10·53-s + 16·55-s − 4·57-s + 4·59-s + 4·61-s + 16·65-s + 4·67-s − 8·71-s − 16·73-s − 11·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 1.03·15-s + 0.917·19-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s + 0.640·39-s + 0.609·43-s − 0.596·45-s + 1.16·47-s + 1.37·53-s + 2.15·55-s − 0.529·57-s + 0.520·59-s + 0.512·61-s + 1.98·65-s + 0.488·67-s − 0.949·71-s − 1.87·73-s − 1.27·75-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.40323234144513235339211494077, −7.07413141125802474536199885825, −5.80093614927017054820846789463, −5.26152249922978530755593786077, −4.56617477562259565309463263802, −3.94097125350035884022119331395, −3.12528878724257586129212671932, −2.31918583208702184792149957570, −0.803156312058767874949497532854, 0, 0.803156312058767874949497532854, 2.31918583208702184792149957570, 3.12528878724257586129212671932, 3.94097125350035884022119331395, 4.56617477562259565309463263802, 5.26152249922978530755593786077, 5.80093614927017054820846789463, 7.07413141125802474536199885825, 7.40323234144513235339211494077

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