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 + 6·13-s + 2·15-s + 2·17-s − 4·19-s + 4·23-s − 25-s + 27-s + 10·29-s − 8·31-s − 6·37-s + 6·39-s + 2·41-s − 4·43-s + 2·45-s + 8·47-s + 2·51-s + 10·53-s − 4·57-s − 12·59-s − 2·61-s + 12·65-s + 12·67-s + 4·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.986·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s + 0.481·69-s + 1.42·71-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

\(L(1)\) \(\approx\) \(3.722696758\)
\(L(\frac12)\) \(\approx\) \(3.722696758\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.82743796807425324682537241028, −6.87106187868184519376645971469, −6.41209090553554583876377982685, −5.68571981664305178650465318250, −5.04049345640250946485687721482, −4.01200981736223031728951406421, −3.47201542341959256724382250812, −2.55307381408546751495005838726, −1.77253258535967865531094218211, −0.953843045434870032500592344220, 0.953843045434870032500592344220, 1.77253258535967865531094218211, 2.55307381408546751495005838726, 3.47201542341959256724382250812, 4.01200981736223031728951406421, 5.04049345640250946485687721482, 5.68571981664305178650465318250, 6.41209090553554583876377982685, 6.87106187868184519376645971469, 7.82743796807425324682537241028

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