Properties

Degree $2$
Conductor $9744$
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 − 2·5-s − 7-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 2·17-s + 4·19-s − 21-s − 25-s + 27-s + 29-s + 8·31-s − 4·33-s + 2·35-s − 10·37-s − 2·39-s − 6·41-s − 12·43-s − 2·45-s + 8·47-s + 49-s + 2·51-s + 6·53-s + 8·55-s + 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.185·29-s + 1.43·31-s − 0.696·33-s + 0.338·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9744\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 29\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9744} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.346826495\)
\(L(\frac12)\) \(\approx\) \(1.346826495\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.75565507138533, −15.71784671458765, −15.54217758450774, −15.31056061328452, −14.28276193801310, −13.77171410655932, −13.38269440202521, −12.46493559152054, −12.13772862172492, −11.58733268574852, −10.68261298767902, −10.12294331509558, −9.713483670878800, −8.832451129225632, −8.141673197776555, −7.810754591937381, −7.145033108211319, −6.525866147282570, −5.394097002126828, −4.984875680107297, −4.067257087542206, −3.274412557731322, −2.858646182060406, −1.821661316773056, −0.5181164277405087, 0.5181164277405087, 1.821661316773056, 2.858646182060406, 3.274412557731322, 4.067257087542206, 4.984875680107297, 5.394097002126828, 6.525866147282570, 7.145033108211319, 7.810754591937381, 8.141673197776555, 8.832451129225632, 9.713483670878800, 10.12294331509558, 10.68261298767902, 11.58733268574852, 12.13772862172492, 12.46493559152054, 13.38269440202521, 13.77171410655932, 14.28276193801310, 15.31056061328452, 15.54217758450774, 15.71784671458765, 16.75565507138533

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