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 + 3·5-s + 9-s + 11-s − 4·13-s + 3·15-s − 4·17-s + 8·23-s + 4·25-s + 27-s + 7·29-s − 11·31-s + 33-s − 4·37-s − 4·39-s + 4·41-s + 2·43-s + 3·45-s + 2·47-s − 4·51-s + 11·53-s + 3·55-s + 7·59-s + 10·61-s − 12·65-s − 10·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.774·15-s − 0.970·17-s + 1.66·23-s + 4/5·25-s + 0.192·27-s + 1.29·29-s − 1.97·31-s + 0.174·33-s − 0.657·37-s − 0.640·39-s + 0.624·41-s + 0.304·43-s + 0.447·45-s + 0.291·47-s − 0.560·51-s + 1.51·53-s + 0.404·55-s + 0.911·59-s + 1.28·61-s − 1.48·65-s − 1.22·67-s + 0.963·69-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

\(L(1)\) \(\approx\) \(3.469394812\)
\(L(\frac12)\) \(\approx\) \(3.469394812\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
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

−7.56353934597226702617837629687, −6.96390425844028393153118206872, −6.51574830709869471820577630900, −5.50723707331241531667270634554, −5.07640809916565415499924409096, −4.24218139198967274955740367903, −3.28332987329258287144683382306, −2.38781677260919829664396697297, −2.03277763608385580162938328125, −0.868644320416121497360422506931, 0.868644320416121497360422506931, 2.03277763608385580162938328125, 2.38781677260919829664396697297, 3.28332987329258287144683382306, 4.24218139198967274955740367903, 5.07640809916565415499924409096, 5.50723707331241531667270634554, 6.51574830709869471820577630900, 6.96390425844028393153118206872, 7.56353934597226702617837629687

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