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 + 0.585·5-s + 9-s − 4.82·11-s + 4.24·13-s + 0.585·15-s − 4.58·17-s − 1.17·19-s − 0.828·23-s − 4.65·25-s + 27-s + 2.82·29-s + 2.82·31-s − 4.82·33-s − 9.65·37-s + 4.24·39-s − 1.75·41-s + 11.3·43-s + 0.585·45-s + 12.4·47-s − 4.58·51-s + 2·53-s − 2.82·55-s − 1.17·57-s + 8.48·59-s − 3.07·61-s + 2.48·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.261·5-s + 0.333·9-s − 1.45·11-s + 1.17·13-s + 0.151·15-s − 1.11·17-s − 0.268·19-s − 0.172·23-s − 0.931·25-s + 0.192·27-s + 0.525·29-s + 0.508·31-s − 0.840·33-s − 1.58·37-s + 0.679·39-s − 0.274·41-s + 1.72·43-s + 0.0873·45-s + 1.82·47-s − 0.642·51-s + 0.274·53-s − 0.381·55-s − 0.155·57-s + 1.10·59-s − 0.393·61-s + 0.308·65-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 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 4.58T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 9.65T + 37T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 3.07T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 6.48T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−7.46498760222159273743634623649, −6.73819596163184691902359302712, −5.92261609293520518516079461205, −5.42270841971141695942205780070, −4.40438842005853880903787914455, −3.90354672070654207970195668352, −2.83912290502608412088540378646, −2.34776637748721021389676289526, −1.37582143557372265510835878909, 0, 1.37582143557372265510835878909, 2.34776637748721021389676289526, 2.83912290502608412088540378646, 3.90354672070654207970195668352, 4.40438842005853880903787914455, 5.42270841971141695942205780070, 5.92261609293520518516079461205, 6.73819596163184691902359302712, 7.46498760222159273743634623649

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