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 + 2·5-s + 9-s + 2·11-s − 13-s − 2·15-s − 19-s − 25-s − 27-s − 4·29-s + 9·31-s − 2·33-s − 3·37-s + 39-s − 10·41-s − 5·43-s + 2·45-s − 6·47-s − 12·53-s + 4·55-s + 57-s + 12·59-s − 10·61-s − 2·65-s + 5·67-s − 6·71-s − 3·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s − 0.229·19-s − 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.61·31-s − 0.348·33-s − 0.493·37-s + 0.160·39-s − 1.56·41-s − 0.762·43-s + 0.298·45-s − 0.875·47-s − 1.64·53-s + 0.539·55-s + 0.132·57-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.610·67-s − 0.712·71-s − 0.351·73-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: $\chi_{9408} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−17.10155622763272, −16.64714013512155, −15.90681377157390, −15.32858233048709, −14.68337937595457, −14.11910273861559, −13.43370523865961, −13.11964797462665, −12.24172765610223, −11.80081288361515, −11.23639173276833, −10.43515792197376, −9.924622433303562, −9.516406237380645, −8.698691530691926, −8.062651556248787, −7.186006579759639, −6.471636780166309, −6.161058902622907, −5.266628103977489, −4.786530211960997, −3.884871804777392, −3.015711805012182, −1.981583624330525, −1.344134596465253, 0, 1.344134596465253, 1.981583624330525, 3.015711805012182, 3.884871804777392, 4.786530211960997, 5.266628103977489, 6.161058902622907, 6.471636780166309, 7.186006579759639, 8.062651556248787, 8.698691530691926, 9.516406237380645, 9.924622433303562, 10.43515792197376, 11.23639173276833, 11.80081288361515, 12.24172765610223, 13.11964797462665, 13.43370523865961, 14.11910273861559, 14.68337937595457, 15.32858233048709, 15.90681377157390, 16.64714013512155, 17.10155622763272

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