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 + 4·11-s − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 25-s − 27-s + 2·29-s − 4·33-s − 6·37-s + 2·39-s − 2·41-s − 4·43-s − 2·45-s − 6·51-s − 6·53-s − 8·55-s + 4·57-s − 12·59-s − 2·61-s + 4·65-s + 4·67-s + 6·73-s + 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.840·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.702·73-s + 0.115·75-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.44376428871628094569671391724, −6.55597781799330496824615300000, −6.19439119239525436838633052918, −5.16773217950502102961008704567, −4.64274308636669756274291953837, −3.75545050483136750219295120519, −3.35852253915733761610986992002, −2.03425588439014896949448393282, −1.08543961772187534196881977330, 0, 1.08543961772187534196881977330, 2.03425588439014896949448393282, 3.35852253915733761610986992002, 3.75545050483136750219295120519, 4.64274308636669756274291953837, 5.16773217950502102961008704567, 6.19439119239525436838633052918, 6.55597781799330496824615300000, 7.44376428871628094569671391724

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