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 − 5-s + 9-s + 5·11-s + 15-s − 4·17-s + 8·19-s + 4·23-s − 4·25-s − 27-s + 5·29-s − 3·31-s − 5·33-s + 4·37-s + 2·43-s − 45-s + 6·47-s + 4·51-s + 9·53-s − 5·55-s − 8·57-s − 11·59-s + 6·61-s − 2·67-s − 4·69-s − 2·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.50·11-s + 0.258·15-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 0.192·27-s + 0.928·29-s − 0.538·31-s − 0.870·33-s + 0.657·37-s + 0.304·43-s − 0.149·45-s + 0.875·47-s + 0.560·51-s + 1.23·53-s − 0.674·55-s − 1.05·57-s − 1.43·59-s + 0.768·61-s − 0.244·67-s − 0.481·69-s − 0.237·71-s + 1.17·73-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

\(L(1)\) \(\approx\) \(1.773597633\)
\(L(\frac12)\) \(\approx\) \(1.773597633\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.44916381549294517320061861042, −7.07914228470291444035944530748, −6.35594085300549375862514986450, −5.70628072806710957422440193877, −4.91446991069822508643151198523, −4.19087776289179662444035478002, −3.62339991614856028528555747353, −2.66626093135574779633322557703, −1.47587214265825167296388406086, −0.71780832588680610562992528143, 0.71780832588680610562992528143, 1.47587214265825167296388406086, 2.66626093135574779633322557703, 3.62339991614856028528555747353, 4.19087776289179662444035478002, 4.91446991069822508643151198523, 5.70628072806710957422440193877, 6.35594085300549375862514986450, 7.07914228470291444035944530748, 7.44916381549294517320061861042

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