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 + 11-s + 15-s − 8·17-s − 4·19-s − 4·23-s − 4·25-s − 27-s + 5·29-s − 7·31-s − 33-s − 8·37-s + 4·41-s + 10·43-s − 45-s − 6·47-s + 8·51-s + 53-s − 55-s + 4·57-s + 9·59-s + 2·61-s + 2·67-s + 4·69-s − 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.258·15-s − 1.94·17-s − 0.917·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 0.928·29-s − 1.25·31-s − 0.174·33-s − 1.31·37-s + 0.624·41-s + 1.52·43-s − 0.149·45-s − 0.875·47-s + 1.12·51-s + 0.137·53-s − 0.134·55-s + 0.529·57-s + 1.17·59-s + 0.256·61-s + 0.244·67-s + 0.481·69-s − 0.712·71-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

\(L(1)\) \(\approx\) \(0.7098454315\)
\(L(\frac12)\) \(\approx\) \(0.7098454315\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.61012898846106706125245696330, −6.91325601420687236280393325998, −6.39277350072801517875123627677, −5.74340808969328145174525955068, −4.85645672870013924898008560586, −4.17908036187774826503955578192, −3.74258617931624501899048022141, −2.44017161313018470386703675470, −1.78536813906594279182960108958, −0.40102795558582767845878793324, 0.40102795558582767845878793324, 1.78536813906594279182960108958, 2.44017161313018470386703675470, 3.74258617931624501899048022141, 4.17908036187774826503955578192, 4.85645672870013924898008560586, 5.74340808969328145174525955068, 6.39277350072801517875123627677, 6.91325601420687236280393325998, 7.61012898846106706125245696330

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