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Properties

Label 24-3332e12-1.1-c0e12-0-1
Degree $24$
Conductor $1.873\times 10^{42}$
Sign $1$
Analytic cond. $447.038$
Root an. cond. $1.28952$
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-s − 13·3-s + 4-s − 13·6-s + 2·7-s + 91·9-s + 8·11-s − 13·12-s − 5·13-s + 2·14-s + 17-s + 91·18-s − 26·21-s + 8·22-s − 2·23-s + 25-s − 5·26-s − 454·27-s + 2·28-s − 2·31-s − 104·33-s + 34-s + 91·36-s + 65·39-s − 26·42-s + 8·44-s − 2·46-s + ⋯
L(s)  = 1  + 2-s − 13·3-s + 4-s − 13·6-s + 2·7-s + 91·9-s + 8·11-s − 13·12-s − 5·13-s + 2·14-s + 17-s + 91·18-s − 26·21-s + 8·22-s − 2·23-s + 25-s − 5·26-s − 454·27-s + 2·28-s − 2·31-s − 104·33-s + 34-s + 91·36-s + 65·39-s − 26·42-s + 8·44-s − 2·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 7^{24} \cdot 17^{12}\)
Sign: $1$
Analytic conductor: \(447.038\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 7^{24} \cdot 17^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04677906070\)
\(L(\frac12)\) \(\approx\) \(0.04677906070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
17 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
good3 \( ( 1 + T )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
13 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
19 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
23 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
67 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
71 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
97 \( ( 1 - T )^{12}( 1 + T )^{12} \)
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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

−2.97362964193008432063377579915, −2.52562505145583689150304974782, −2.46079896182330565401828073094, −2.36006277633540687133552307198, −2.33105082181338681546076180988, −2.27964117513428697604530071209, −2.13777570569629727145661093096, −2.09130813610060044664796470791, −1.76624598299137482936908931135, −1.68784286700694512684485983916, −1.65034884004911691028303235707, −1.62410113246871487747156719430, −1.60605356429206217793828883695, −1.57258342611680514737277280315, −1.46237620828986753753954683686, −1.44193082633587682056472683209, −1.22566605082284894282357981892, −1.22187240350877193597530093889, −1.20428093689085394173066419885, −0.919696082950570671571896365228, −0.818350586665259534052658915550, −0.77137282632510650982950752861, −0.64355780734717436148257273164, −0.38194429820159092299038854588, −0.26391246481764914994674035999, 0.26391246481764914994674035999, 0.38194429820159092299038854588, 0.64355780734717436148257273164, 0.77137282632510650982950752861, 0.818350586665259534052658915550, 0.919696082950570671571896365228, 1.20428093689085394173066419885, 1.22187240350877193597530093889, 1.22566605082284894282357981892, 1.44193082633587682056472683209, 1.46237620828986753753954683686, 1.57258342611680514737277280315, 1.60605356429206217793828883695, 1.62410113246871487747156719430, 1.65034884004911691028303235707, 1.68784286700694512684485983916, 1.76624598299137482936908931135, 2.09130813610060044664796470791, 2.13777570569629727145661093096, 2.27964117513428697604530071209, 2.33105082181338681546076180988, 2.36006277633540687133552307198, 2.46079896182330565401828073094, 2.52562505145583689150304974782, 2.97362964193008432063377579915

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

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