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 + 9-s − 5.65·11-s + 5.65·13-s − 5.65·17-s + 4·19-s + 5.65·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s + 5.65·33-s − 2·37-s − 5.65·39-s + 5.65·41-s − 8·47-s + 5.65·51-s + 2·53-s − 4·57-s + 4·59-s + 5.65·61-s + 11.3·67-s − 5.65·69-s − 5.65·71-s + 11.3·73-s + 5·75-s − 11.3·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.333·9-s − 1.70·11-s + 1.56·13-s − 1.37·17-s + 0.917·19-s + 1.17·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.984·33-s − 0.328·37-s − 0.905·39-s + 0.883·41-s − 1.16·47-s + 0.792·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s + 0.724·61-s + 1.38·67-s − 0.681·69-s − 0.671·71-s + 1.32·73-s + 0.577·75-s − 1.27·79-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: \(1\)
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 \( 1 + 5T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 5.65T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.21728830975177467499149622850, −6.74729704993067497746749882969, −5.83199696764183515886776601216, −5.41251345812635484926475327871, −4.70578945366403615049158522883, −3.86397682604065271570758608836, −3.04184520970393234090360317844, −2.17069110891621622709579115728, −1.10927131514791689287240561276, 0, 1.10927131514791689287240561276, 2.17069110891621622709579115728, 3.04184520970393234090360317844, 3.86397682604065271570758608836, 4.70578945366403615049158522883, 5.41251345812635484926475327871, 5.83199696764183515886776601216, 6.74729704993067497746749882969, 7.21728830975177467499149622850

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