Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.04·5-s + 9-s + 5.93·11-s + 0.888·13-s + 1.04·15-s + 4.51·17-s + 3.47·19-s + 1.93·23-s − 3.91·25-s + 27-s + 8.38·29-s − 5.13·31-s + 5.93·33-s − 5.65·37-s + 0.888·39-s + 8.39·41-s − 0.743·43-s + 1.04·45-s + 4.21·47-s + 4.51·51-s + 6.91·53-s + 6.18·55-s + 3.47·57-s + 5.43·59-s − 10.9·61-s + 0.926·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.466·5-s + 0.333·9-s + 1.78·11-s + 0.246·13-s + 0.269·15-s + 1.09·17-s + 0.797·19-s + 0.402·23-s − 0.782·25-s + 0.192·27-s + 1.55·29-s − 0.921·31-s + 1.03·33-s − 0.929·37-s + 0.142·39-s + 1.31·41-s − 0.113·43-s + 0.155·45-s + 0.615·47-s + 0.632·51-s + 0.949·53-s + 0.833·55-s + 0.460·57-s + 0.708·59-s − 1.40·61-s + 0.114·65-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: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.878592148\)
\(L(\frac12)\) \(\approx\) \(3.878592148\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.04T + 5T^{2} \)
11 \( 1 - 5.93T + 11T^{2} \)
13 \( 1 - 0.888T + 13T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 - 3.47T + 19T^{2} \)
23 \( 1 - 1.93T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 8.39T + 41T^{2} \)
43 \( 1 + 0.743T + 43T^{2} \)
47 \( 1 - 4.21T + 47T^{2} \)
53 \( 1 - 6.91T + 53T^{2} \)
59 \( 1 - 5.43T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 6.20T + 67T^{2} \)
71 \( 1 + 3.72T + 71T^{2} \)
73 \( 1 + 3.49T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 2.94T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.49276209603864326455532967754, −7.22945095052265514866284580562, −6.24078608074584322523755916095, −5.83157370151350532377696708582, −4.89743906608515189752734151749, −4.02413105192893999837081609166, −3.48456388054080826628237662221, −2.67954144253839376605831423293, −1.59058360785738361294948635896, −1.03955488540382184635414131454, 1.03955488540382184635414131454, 1.59058360785738361294948635896, 2.67954144253839376605831423293, 3.48456388054080826628237662221, 4.02413105192893999837081609166, 4.89743906608515189752734151749, 5.83157370151350532377696708582, 6.24078608074584322523755916095, 7.22945095052265514866284580562, 7.49276209603864326455532967754

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