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 + 2·13-s + 2·15-s − 2·17-s − 4·19-s − 25-s + 27-s − 6·29-s − 6·37-s + 2·39-s + 6·41-s + 8·43-s + 2·45-s + 8·47-s − 2·51-s − 6·53-s − 4·57-s + 12·59-s + 10·61-s + 4·65-s + 16·67-s + 8·71-s + 6·73-s − 75-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s + 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.95·67-s + 0.949·71-s + 0.702·73-s − 0.115·75-s − 0.900·79-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

\(L(1)\) \(\approx\) \(3.149383564\)
\(L(\frac12)\) \(\approx\) \(3.149383564\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.72603143607158752044686741875, −7.01457138950833382896839978793, −6.30733389772304530213003727973, −5.74071340127034582561549161433, −4.99008942844597342041954417920, −4.02541501977557652775434352579, −3.54527166050418800052699168253, −2.20363931114562236079736795767, −2.15422604112595950745863621955, −0.820749216643845756760176902569, 0.820749216643845756760176902569, 2.15422604112595950745863621955, 2.20363931114562236079736795767, 3.54527166050418800052699168253, 4.02541501977557652775434352579, 4.99008942844597342041954417920, 5.74071340127034582561549161433, 6.30733389772304530213003727973, 7.01457138950833382896839978793, 7.72603143607158752044686741875

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