Properties

Degree $2$
Conductor $159936$
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 + 9-s + 4·11-s − 2·13-s − 2·15-s − 17-s − 4·19-s − 8·23-s − 25-s + 27-s − 6·29-s + 4·33-s + 2·37-s − 2·39-s − 10·41-s − 4·43-s − 2·45-s − 51-s − 6·53-s − 8·55-s − 4·57-s + 4·59-s + 6·61-s + 4·65-s − 12·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.140·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.768·61-s + 0.496·65-s − 1.46·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{159936} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6709003781\)
\(L(\frac12)\) \(\approx\) \(0.6709003781\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.29273508695369, −12.78733525102909, −12.19165047442238, −11.95007370957775, −11.44423653188706, −11.04504768959321, −10.28936759802908, −9.910346884447297, −9.434843659548820, −8.870725311589705, −8.455793501019039, −7.943563528088051, −7.579517908967650, −7.003528340011052, −6.409169144199781, −6.132067334046194, −5.223416918066153, −4.687646281970889, −4.031359131220226, −3.779679362749343, −3.352501520977433, −2.357503971821026, −1.981763213826892, −1.315130389070702, −0.2246248920318965, 0.2246248920318965, 1.315130389070702, 1.981763213826892, 2.357503971821026, 3.352501520977433, 3.779679362749343, 4.031359131220226, 4.687646281970889, 5.223416918066153, 6.132067334046194, 6.409169144199781, 7.003528340011052, 7.579517908967650, 7.943563528088051, 8.455793501019039, 8.870725311589705, 9.434843659548820, 9.910346884447297, 10.28936759802908, 11.04504768959321, 11.44423653188706, 11.95007370957775, 12.19165047442238, 12.78733525102909, 13.29273508695369

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