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Properties

Label 24-63e24-1.1-c0e12-0-3
Degree $24$
Conductor $1.528\times 10^{43}$
Sign $1$
Analytic cond. $3648.03$
Root an. cond. $1.40740$
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  − 2·4-s + 7-s − 13-s + 16-s + 2·19-s + 25-s − 2·28-s + 2·31-s + 13·37-s − 43-s + 49-s + 2·52-s − 5·61-s + 2·67-s + 2·73-s − 4·76-s + 2·79-s − 91-s − 97-s − 2·100-s − 103-s − 109-s + 112-s + 121-s − 4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2·4-s + 7-s − 13-s + 16-s + 2·19-s + 25-s − 2·28-s + 2·31-s + 13·37-s − 43-s + 49-s + 2·52-s − 5·61-s + 2·67-s + 2·73-s − 4·76-s + 2·79-s − 91-s − 97-s − 2·100-s − 103-s − 109-s + 112-s + 121-s − 4·124-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{48} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(3648.03\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{48} \cdot 7^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.133754862\)
\(L(\frac12)\) \(\approx\) \(2.133754862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
37 \( ( 1 - T )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
67 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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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

−2.72343705761253584003044028453, −2.71946151405560196917343636805, −2.51063234929655359585840749181, −2.49999096208060386781603128929, −2.43791601987764839432208632350, −2.34536870133876734970716775262, −2.28355826536084198621502043544, −2.23207445564970490958627529114, −2.21028911304560283433176797995, −2.15929147613364424943647711626, −2.13523697470875816880840198875, −1.86944991938733577122759715191, −1.71772003279302345609541340712, −1.41443713471656574775465933126, −1.27938204874974313350473804496, −1.24505604772280292092283920774, −1.22685535397770199833614127525, −1.20787875589515877330683899581, −1.18454278821053759701428743417, −1.17508361479285033514065077375, −0.954004844528955802905497898649, −0.815110193680977345320228354288, −0.75232140979003004764444373996, −0.48037024905993260663904030957, −0.31066957903571360051885695811, 0.31066957903571360051885695811, 0.48037024905993260663904030957, 0.75232140979003004764444373996, 0.815110193680977345320228354288, 0.954004844528955802905497898649, 1.17508361479285033514065077375, 1.18454278821053759701428743417, 1.20787875589515877330683899581, 1.22685535397770199833614127525, 1.24505604772280292092283920774, 1.27938204874974313350473804496, 1.41443713471656574775465933126, 1.71772003279302345609541340712, 1.86944991938733577122759715191, 2.13523697470875816880840198875, 2.15929147613364424943647711626, 2.21028911304560283433176797995, 2.23207445564970490958627529114, 2.28355826536084198621502043544, 2.34536870133876734970716775262, 2.43791601987764839432208632350, 2.49999096208060386781603128929, 2.51063234929655359585840749181, 2.71946151405560196917343636805, 2.72343705761253584003044028453

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

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