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Properties

Label 24-3332e12-1.1-c0e12-0-0
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  + 2·2-s + 4-s − 5·9-s − 10·13-s − 2·17-s − 10·18-s − 2·25-s − 20·26-s − 4·34-s − 5·36-s + 49-s − 4·50-s − 10·52-s − 4·53-s − 2·68-s + 15·81-s + 4·89-s + 2·98-s − 2·100-s + 4·101-s − 8·106-s + 50·117-s − 5·121-s + 127-s − 2·128-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 5·9-s − 10·13-s − 2·17-s − 10·18-s − 2·25-s − 20·26-s − 4·34-s − 5·36-s + 49-s − 4·50-s − 10·52-s − 4·53-s − 2·68-s + 15·81-s + 4·89-s + 2·98-s − 2·100-s + 4·101-s − 8·106-s + 50·117-s − 5·121-s + 127-s − 2·128-s + 131-s + 137-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

\(L(\frac{1}{2})\) \(\approx\) \(0.02721248609\)
\(L(\frac12)\) \(\approx\) \(0.02721248609\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
good3 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
11 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
13 \( ( 1 + T )^{12}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
19 \( ( 1 - T )^{12}( 1 + T )^{12} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
67 \( ( 1 - T )^{12}( 1 + T )^{12} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4} \)
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

−2.80238158492081189414608585623, −2.73343573491181054499224866888, −2.61803483441737538556162051507, −2.51101119212846296005406007100, −2.50349385747068693840208406740, −2.43281982168762132099449891987, −2.34767431397432780283828937594, −2.34580724889632683212878475329, −2.33110968577581435849534891327, −2.31574799004138625653225051349, −2.27245219337206432788446173304, −1.95040540858018553602810501150, −1.90841375903739943625478683330, −1.86710255932552172016212034657, −1.76809461783829333689950856074, −1.68636441992371162168697999859, −1.47588353428623989378900455585, −1.40432744376348382392468374215, −1.15215486302230537003789172583, −1.12726891505975528184913036096, −0.71123287276723704805990679413, −0.64294837586593912064272987617, −0.37513215123562130472116971132, −0.27448086789833278714479463189, −0.10767528396247352350189559937, 0.10767528396247352350189559937, 0.27448086789833278714479463189, 0.37513215123562130472116971132, 0.64294837586593912064272987617, 0.71123287276723704805990679413, 1.12726891505975528184913036096, 1.15215486302230537003789172583, 1.40432744376348382392468374215, 1.47588353428623989378900455585, 1.68636441992371162168697999859, 1.76809461783829333689950856074, 1.86710255932552172016212034657, 1.90841375903739943625478683330, 1.95040540858018553602810501150, 2.27245219337206432788446173304, 2.31574799004138625653225051349, 2.33110968577581435849534891327, 2.34580724889632683212878475329, 2.34767431397432780283828937594, 2.43281982168762132099449891987, 2.50349385747068693840208406740, 2.51101119212846296005406007100, 2.61803483441737538556162051507, 2.73343573491181054499224866888, 2.80238158492081189414608585623

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

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