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 − 4·5-s + 9-s − 2·11-s − 2·13-s + 4·15-s + 4·19-s + 6·23-s + 11·25-s − 27-s + 10·29-s − 8·31-s + 2·33-s − 10·37-s + 2·39-s + 4·41-s − 8·43-s − 4·45-s − 4·47-s − 10·53-s + 8·55-s − 4·57-s − 8·59-s − 6·61-s + 8·65-s + 4·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.917·19-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.348·33-s − 1.64·37-s + 0.320·39-s + 0.624·41-s − 1.21·43-s − 0.596·45-s − 0.583·47-s − 1.37·53-s + 1.07·55-s − 0.529·57-s − 1.04·59-s − 0.768·61-s + 0.992·65-s + 0.488·67-s − 0.722·69-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

\(L(1)\) \(\approx\) \(0.5449962001\)
\(L(\frac12)\) \(\approx\) \(0.5449962001\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.59558207791424671328776436287, −7.13323770971562337547616543648, −6.56474797803428897498590792204, −5.41823570283133765913163545172, −4.87504963699550784963858338089, −4.38884873889324384252347421918, −3.29854605998433546554677404944, −3.01539426879637264959862099286, −1.47672196729151907891865080856, −0.37845498970653377118116825344, 0.37845498970653377118116825344, 1.47672196729151907891865080856, 3.01539426879637264959862099286, 3.29854605998433546554677404944, 4.38884873889324384252347421918, 4.87504963699550784963858338089, 5.41823570283133765913163545172, 6.56474797803428897498590792204, 7.13323770971562337547616543648, 7.59558207791424671328776436287

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