Properties

Degree $2$
Conductor $56784$
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 − 7-s + 9-s − 4·11-s + 2·15-s + 2·17-s − 4·19-s − 21-s − 8·23-s − 25-s + 27-s − 2·29-s − 4·33-s − 2·35-s + 10·37-s + 6·41-s + 4·43-s + 2·45-s + 49-s + 2·51-s + 6·53-s − 8·55-s − 4·57-s + 4·59-s + 6·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.338·35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56784\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{56784} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 56784,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.59438932055840, −14.08725250358439, −13.60377060262652, −13.04246825143206, −12.89456052440251, −12.23803673443663, −11.59257935466564, −10.91530942038541, −10.30923635393299, −10.05107122828424, −9.516367698983017, −9.075009840442554, −8.260571291149977, −7.954029111488519, −7.444081447076526, −6.689138293707602, −5.992110328407495, −5.768901925720514, −5.086755356093434, −4.198169242786379, −3.874195798854997, −2.921366143688300, −2.324066926937505, −2.082236677656441, −1.003702378272600, 0, 1.003702378272600, 2.082236677656441, 2.324066926937505, 2.921366143688300, 3.874195798854997, 4.198169242786379, 5.086755356093434, 5.768901925720514, 5.992110328407495, 6.689138293707602, 7.444081447076526, 7.954029111488519, 8.260571291149977, 9.075009840442554, 9.516367698983017, 10.05107122828424, 10.30923635393299, 10.91530942038541, 11.59257935466564, 12.23803673443663, 12.89456052440251, 13.04246825143206, 13.60377060262652, 14.08725250358439, 14.59438932055840

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