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 − 2·5-s + 9-s + 4·11-s + 6·13-s + 2·15-s − 2·17-s − 4·19-s + 8·23-s − 25-s − 27-s + 2·29-s − 4·33-s + 10·37-s − 6·39-s + 6·41-s + 4·43-s − 2·45-s + 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s + 6·61-s − 12·65-s − 4·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 1.64·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.768·61-s − 1.48·65-s − 0.488·67-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

\(L(1)\) \(\approx\) \(1.685869771\)
\(L(\frac12)\) \(\approx\) \(1.685869771\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.68543261494553534250932657704, −6.86752699098244453021865394927, −6.36638552476027123983992946496, −5.85369460716372440911607899279, −4.78355606415300066608847946606, −4.10418829003632411248203207487, −3.75326864427413445495438802807, −2.69953192309157863520716245553, −1.41819475066661361165135718507, −0.71588246997227816408272652354, 0.71588246997227816408272652354, 1.41819475066661361165135718507, 2.69953192309157863520716245553, 3.75326864427413445495438802807, 4.10418829003632411248203207487, 4.78355606415300066608847946606, 5.85369460716372440911607899279, 6.36638552476027123983992946496, 6.86752699098244453021865394927, 7.68543261494553534250932657704

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