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 + 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: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276745168\)
\(L(\frac12)\) \(\approx\) \(1.276745168\)
\(L(\frac{3}{2})\) 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

−7.77674693440643697193166623684, −6.75566082151390302248751202000, −6.36997002446181083239530639264, −5.70703861556459633919414942692, −5.06652079482889582156900438487, −4.10693190428278308828568760532, −3.70689379851316482669561445604, −2.38957193684007609002169207366, −1.80096783372793063424528168776, −0.54567255929743760849109150762, 0.54567255929743760849109150762, 1.80096783372793063424528168776, 2.38957193684007609002169207366, 3.70689379851316482669561445604, 4.10693190428278308828568760532, 5.06652079482889582156900438487, 5.70703861556459633919414942692, 6.36997002446181083239530639264, 6.75566082151390302248751202000, 7.77674693440643697193166623684

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