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 + 0.585·5-s + 9-s − 0.828·11-s − 1.41·13-s − 0.585·15-s − 2.24·17-s − 6.82·19-s − 4.82·23-s − 4.65·25-s − 27-s − 8.48·29-s + 5.17·31-s + 0.828·33-s − 1.65·37-s + 1.41·39-s + 0.585·41-s − 8·43-s + 0.585·45-s + 6.82·47-s + 2.24·51-s + 13.3·53-s − 0.485·55-s + 6.82·57-s − 5.17·59-s + 13.8·61-s − 0.828·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.261·5-s + 0.333·9-s − 0.249·11-s − 0.392·13-s − 0.151·15-s − 0.543·17-s − 1.56·19-s − 1.00·23-s − 0.931·25-s − 0.192·27-s − 1.57·29-s + 0.928·31-s + 0.144·33-s − 0.272·37-s + 0.226·39-s + 0.0914·41-s − 1.21·43-s + 0.0873·45-s + 0.996·47-s + 0.314·51-s + 1.82·53-s − 0.0654·55-s + 0.904·57-s − 0.673·59-s + 1.77·61-s − 0.102·65-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

\(L(1)\) \(\approx\) \(0.9289317297\)
\(L(\frac12)\) \(\approx\) \(0.9289317297\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 5.17T + 31T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 0.828T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 7.75T + 97T^{2} \)
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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

−7.59754952493918638532494705160, −6.98749367846939955155565705517, −6.19337992088851168134332544181, −5.79716131908504674783806117743, −4.96802595292866187351412874698, −4.24688682768405466642746601112, −3.63303687257312336016518181841, −2.30101040335878366305906454952, −1.92645126436223169599472639519, −0.45180679034758887988236174591, 0.45180679034758887988236174591, 1.92645126436223169599472639519, 2.30101040335878366305906454952, 3.63303687257312336016518181841, 4.24688682768405466642746601112, 4.96802595292866187351412874698, 5.79716131908504674783806117743, 6.19337992088851168134332544181, 6.98749367846939955155565705517, 7.59754952493918638532494705160

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