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 + 9-s + 2·11-s − 2·13-s − 4·17-s − 4·19-s − 6·23-s − 5·25-s − 27-s + 2·29-s − 2·33-s + 6·37-s + 2·39-s − 8·41-s + 8·43-s + 4·47-s + 4·51-s + 6·53-s + 4·57-s − 14·61-s − 4·67-s + 6·69-s − 2·71-s + 2·73-s + 5·75-s + 4·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.986·37-s + 0.320·39-s − 1.24·41-s + 1.21·43-s + 0.583·47-s + 0.560·51-s + 0.824·53-s + 0.529·57-s − 1.79·61-s − 0.488·67-s + 0.722·69-s − 0.237·71-s + 0.234·73-s + 0.577·75-s + 0.450·79-s + 1/9·81-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: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066245406\)
\(L(\frac12)\) \(\approx\) \(1.066245406\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.82264111496370, −16.25346773184272, −15.49185646730644, −15.19108294576054, −14.42502943328364, −13.78112702886298, −13.31261601165095, −12.48829811813109, −12.06642051781293, −11.54540992965716, −10.85362138453959, −10.30707455200945, −9.667082057933677, −9.040153310068623, −8.350980683878431, −7.603589303538587, −6.986495475147786, −6.095088282855529, −5.992110987245620, −4.779933608211945, −4.341445125183391, −3.627748558555836, −2.414037821072251, −1.797379779930245, −0.4955289902294057, 0.4955289902294057, 1.797379779930245, 2.414037821072251, 3.627748558555836, 4.341445125183391, 4.779933608211945, 5.992110987245620, 6.095088282855529, 6.986495475147786, 7.603589303538587, 8.350980683878431, 9.040153310068623, 9.667082057933677, 10.30707455200945, 10.85362138453959, 11.54540992965716, 12.06642051781293, 12.48829811813109, 13.31261601165095, 13.78112702886298, 14.42502943328364, 15.19108294576054, 15.49185646730644, 16.25346773184272, 16.82264111496370

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