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 + 9-s − 2·11-s − 5·13-s − 2·17-s − 3·19-s + 2·23-s − 5·25-s − 27-s − 8·29-s − 31-s + 2·33-s + 5·37-s + 5·39-s + 2·41-s − 7·43-s − 8·47-s + 2·51-s + 2·53-s + 3·57-s − 10·59-s + 2·61-s + 11·67-s − 2·69-s + 12·71-s − 3·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.38·13-s − 0.485·17-s − 0.688·19-s + 0.417·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s + 0.348·33-s + 0.821·37-s + 0.800·39-s + 0.312·41-s − 1.06·43-s − 1.16·47-s + 0.280·51-s + 0.274·53-s + 0.397·57-s − 1.30·59-s + 0.256·61-s + 1.34·67-s − 0.240·69-s + 1.42·71-s − 0.351·73-s + 0.577·75-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: $\chi_{9408} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6673869601\)
\(L(\frac12)\) \(\approx\) \(0.6673869601\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.61765877985039, −16.42214724158907, −15.48392450054813, −15.00520902949204, −14.68509145192549, −13.66352241297874, −13.23018893359981, −12.60407525521776, −12.13976846962930, −11.36721377308434, −10.95070339078524, −10.29850712595118, −9.520591711002331, −9.301033612412661, −8.028129315283700, −7.831419254985963, −6.902505936772083, −6.440577806587178, −5.497233761230857, −5.064039042144205, −4.334929052575655, −3.513888439621069, −2.443913516389370, −1.849501944124714, −0.3819567857560517, 0.3819567857560517, 1.849501944124714, 2.443913516389370, 3.513888439621069, 4.334929052575655, 5.064039042144205, 5.497233761230857, 6.440577806587178, 6.902505936772083, 7.831419254985963, 8.028129315283700, 9.301033612412661, 9.520591711002331, 10.29850712595118, 10.95070339078524, 11.36721377308434, 12.13976846962930, 12.60407525521776, 13.23018893359981, 13.66352241297874, 14.68509145192549, 15.00520902949204, 15.48392450054813, 16.42214724158907, 16.61765877985039

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