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 − 2.74·5-s + 9-s − 1.54·11-s − 6.03·13-s + 2.74·15-s + 7.49·17-s + 6.03·19-s − 7.49·23-s + 2.54·25-s − 27-s − 1.25·29-s + 5.29·31-s + 1.54·33-s − 4.94·37-s + 6.03·39-s − 5.08·41-s − 3.45·43-s − 2.74·45-s + 9.49·47-s − 7.49·51-s + 3.83·53-s + 4.23·55-s − 6.03·57-s − 5.54·59-s + 14.5·61-s + 16.5·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.22·5-s + 0.333·9-s − 0.465·11-s − 1.67·13-s + 0.709·15-s + 1.81·17-s + 1.38·19-s − 1.56·23-s + 0.508·25-s − 0.192·27-s − 0.232·29-s + 0.950·31-s + 0.268·33-s − 0.813·37-s + 0.966·39-s − 0.794·41-s − 0.527·43-s − 0.409·45-s + 1.38·47-s − 1.04·51-s + 0.526·53-s + 0.571·55-s − 0.799·57-s − 0.721·59-s + 1.86·61-s + 2.05·65-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 \( 1 + 2.74T + 5T^{2} \)
11 \( 1 + 1.54T + 11T^{2} \)
13 \( 1 + 6.03T + 13T^{2} \)
17 \( 1 - 7.49T + 17T^{2} \)
19 \( 1 - 6.03T + 19T^{2} \)
23 \( 1 + 7.49T + 23T^{2} \)
29 \( 1 + 1.25T + 29T^{2} \)
31 \( 1 - 5.29T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 - 9.49T + 47T^{2} \)
53 \( 1 - 3.83T + 53T^{2} \)
59 \( 1 + 5.54T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 - 4.03T + 67T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 7.79T + 79T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + 9.49T + 89T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−7.45437395031443800633494234256, −6.92409048234475903406407757597, −5.78946789290990515161331899952, −5.29618087820360876087910651744, −4.68055038376497631588766297219, −3.79714993411567762703814108660, −3.21401740570025089886595575227, −2.21455461330791820123463672040, −0.927590678553471684916812242098, 0, 0.927590678553471684916812242098, 2.21455461330791820123463672040, 3.21401740570025089886595575227, 3.79714993411567762703814108660, 4.68055038376497631588766297219, 5.29618087820360876087910651744, 5.78946789290990515161331899952, 6.92409048234475903406407757597, 7.45437395031443800633494234256

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