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Properties

Label 24-516e12-1.1-c0e12-0-0
Degree $24$
Conductor $3.563\times 10^{32}$
Sign $1$
Analytic cond. $8.50514\times 10^{-8}$
Root an. cond. $0.507461$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 13-s + 2·19-s − 21-s + 25-s − 8·31-s − 37-s − 39-s − 2·43-s + 2·57-s + 2·61-s − 63-s − 67-s − 73-s + 75-s + 2·79-s + 91-s − 8·93-s − 4·97-s + 2·103-s − 5·109-s − 111-s − 117-s − 2·121-s + 127-s + ⋯
L(s)  = 1  + 3-s − 7-s + 9-s − 13-s + 2·19-s − 21-s + 25-s − 8·31-s − 37-s − 39-s − 2·43-s + 2·57-s + 2·61-s − 63-s − 67-s − 73-s + 75-s + 2·79-s + 91-s − 8·93-s − 4·97-s + 2·103-s − 5·109-s − 111-s − 117-s − 2·121-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 43^{12}\)
Sign: $1$
Analytic conductor: \(8.50514\times 10^{-8}\)
Root analytic conductor: \(0.507461\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{12} \cdot 43^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1977685312\)
\(L(\frac12)\) \(\approx\) \(0.1977685312\)
\(L(1)\) 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 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
good5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
31 \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.90757964842217630161808982958, −3.73332107058017712284640710182, −3.69552046755138746521138911178, −3.45786655197804985393016348676, −3.38132964660118330180222288362, −3.36987345724300251344288563469, −3.32409867507627423856449700311, −3.22862322368351814404709709020, −3.05831217425121931089042612341, −2.92462146280134150623204781257, −2.91050624731487007608887636814, −2.84316592871393154621194288034, −2.71884952372058119802461983320, −2.67884973764040563079755774556, −2.17308982422574918136164378548, −2.07468536028928193422553950663, −1.96397171664092540563168519437, −1.95658466571377540812193600818, −1.95546612996976892046575659105, −1.92942650214100843817104342083, −1.64983617087780525183719529730, −1.46530145631841582732821105113, −1.39428120973444292810646337482, −1.03385539379676218284006409999, −0.839246908149332121384391368755, 0.839246908149332121384391368755, 1.03385539379676218284006409999, 1.39428120973444292810646337482, 1.46530145631841582732821105113, 1.64983617087780525183719529730, 1.92942650214100843817104342083, 1.95546612996976892046575659105, 1.95658466571377540812193600818, 1.96397171664092540563168519437, 2.07468536028928193422553950663, 2.17308982422574918136164378548, 2.67884973764040563079755774556, 2.71884952372058119802461983320, 2.84316592871393154621194288034, 2.91050624731487007608887636814, 2.92462146280134150623204781257, 3.05831217425121931089042612341, 3.22862322368351814404709709020, 3.32409867507627423856449700311, 3.36987345724300251344288563469, 3.38132964660118330180222288362, 3.45786655197804985393016348676, 3.69552046755138746521138911178, 3.73332107058017712284640710182, 3.90757964842217630161808982958

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

Plot not available for L-functions of degree greater than 10.