Properties

Degree $2$
Conductor $9408$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 9-s + 3·11-s + 4·13-s − 3·15-s − 4·19-s + 4·25-s + 27-s − 9·29-s + 31-s + 3·33-s − 8·37-s + 4·39-s − 10·43-s − 3·45-s + 6·47-s + 3·53-s − 9·55-s − 4·57-s + 3·59-s + 10·61-s − 12·65-s − 10·67-s + 6·71-s + 2·73-s + 4·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.904·11-s + 1.10·13-s − 0.774·15-s − 0.917·19-s + 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.179·31-s + 0.522·33-s − 1.31·37-s + 0.640·39-s − 1.52·43-s − 0.447·45-s + 0.875·47-s + 0.412·53-s − 1.21·55-s − 0.529·57-s + 0.390·59-s + 1.28·61-s − 1.48·65-s − 1.22·67-s + 0.712·71-s + 0.234·73-s + 0.461·75-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: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.37517649142723777905964103636, −6.85519931833545867984665880296, −6.14230795966911920424058957647, −5.21071858021618780124569331236, −4.23008914207635545196170272120, −3.77255066875771816333965788661, −3.38402345370024216077026073627, −2.15616200235712601407212146232, −1.25649655794499947471059629921, 0, 1.25649655794499947471059629921, 2.15616200235712601407212146232, 3.38402345370024216077026073627, 3.77255066875771816333965788661, 4.23008914207635545196170272120, 5.21071858021618780124569331236, 6.14230795966911920424058957647, 6.85519931833545867984665880296, 7.37517649142723777905964103636

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