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 + 2·11-s − 5.41·13-s + 0.585·15-s + 6.24·17-s − 2.82·19-s + 3.65·23-s − 4.65·25-s − 27-s + 1.17·29-s − 6.82·31-s − 2·33-s + 4·37-s + 5.41·39-s − 2.24·41-s + 5.65·43-s − 0.585·45-s + 2.82·47-s − 6.24·51-s + 2·53-s − 1.17·55-s + 2.82·57-s + 6.82·59-s − 3.75·61-s + 3.17·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.261·5-s + 0.333·9-s + 0.603·11-s − 1.50·13-s + 0.151·15-s + 1.51·17-s − 0.648·19-s + 0.762·23-s − 0.931·25-s − 0.192·27-s + 0.217·29-s − 1.22·31-s − 0.348·33-s + 0.657·37-s + 0.866·39-s − 0.350·41-s + 0.862·43-s − 0.0873·45-s + 0.412·47-s − 0.874·51-s + 0.274·53-s − 0.157·55-s + 0.374·57-s + 0.888·59-s − 0.481·61-s + 0.393·65-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 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 + 3.75T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 5.89T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 5.41T + 97T^{2} \)
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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

−7.44580358884736283410009563597, −6.74289247590207632968240427264, −5.87595854259471568082728533687, −5.38897811734563400694683718008, −4.59039069376921842145555086532, −3.94490002001419325246999053548, −3.07834121522320874497585029493, −2.12944986438998220534557064232, −1.10897008304573366247725079464, 0, 1.10897008304573366247725079464, 2.12944986438998220534557064232, 3.07834121522320874497585029493, 3.94490002001419325246999053548, 4.59039069376921842145555086532, 5.38897811734563400694683718008, 5.87595854259471568082728533687, 6.74289247590207632968240427264, 7.44580358884736283410009563597

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