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Properties

Label 24-860e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.637\times 10^{35}$
Sign $1$
Analytic cond. $3.90721\times 10^{-5}$
Root an. cond. $0.655130$
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 − 8·3-s + 4-s + 5-s + 16·6-s + 2·7-s + 28·9-s − 2·10-s − 8·12-s − 4·14-s − 8·15-s − 56·18-s + 20-s − 16·21-s − 5·23-s + 25-s − 50·27-s + 2·28-s − 29-s + 16·30-s + 2·35-s + 28·36-s + 2·41-s + 32·42-s + 43-s + 28·45-s + 10·46-s + ⋯
L(s)  = 1  − 2·2-s − 8·3-s + 4-s + 5-s + 16·6-s + 2·7-s + 28·9-s − 2·10-s − 8·12-s − 4·14-s − 8·15-s − 56·18-s + 20-s − 16·21-s − 5·23-s + 25-s − 50·27-s + 2·28-s − 29-s + 16·30-s + 2·35-s + 28·36-s + 2·41-s + 32·42-s + 43-s + 28·45-s + 10·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 43^{12}\)
Sign: $1$
Analytic conductor: \(3.90721\times 10^{-5}\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 43^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002889783337\)
\(L(\frac12)\) \(\approx\) \(0.002889783337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
43 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
good3 \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
23 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
37 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
79 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
83 \( ( 1 - T )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−3.72773915153050512239087083157, −3.61632512735925350678203945908, −3.33500685196048792795422182619, −3.30960808146773401832587090870, −3.17767025423862504820908907312, −3.04617421719695096496010427456, −2.71404712958156936146360412471, −2.68643150815907301793171887506, −2.58940051035748564457691910400, −2.48028327108556910088495356129, −2.44670332525210685146141515288, −2.43881464933894922051080562381, −2.33637641974289056106423379297, −2.09474073334802177763162174768, −2.05989637746259023555161580680, −1.66631879862142323422191767919, −1.65248614080162205480690007235, −1.63678890331825606922823717541, −1.47998213900589971687341587847, −1.22924584790020120435310953857, −0.903918708845000453669848585957, −0.875618836592400173533502874536, −0.68269130586316330506838704618, −0.59728879185810081389418717149, −0.44581868778082234416969175701, 0.44581868778082234416969175701, 0.59728879185810081389418717149, 0.68269130586316330506838704618, 0.875618836592400173533502874536, 0.903918708845000453669848585957, 1.22924584790020120435310953857, 1.47998213900589971687341587847, 1.63678890331825606922823717541, 1.65248614080162205480690007235, 1.66631879862142323422191767919, 2.05989637746259023555161580680, 2.09474073334802177763162174768, 2.33637641974289056106423379297, 2.43881464933894922051080562381, 2.44670332525210685146141515288, 2.48028327108556910088495356129, 2.58940051035748564457691910400, 2.68643150815907301793171887506, 2.71404712958156936146360412471, 3.04617421719695096496010427456, 3.17767025423862504820908907312, 3.30960808146773401832587090870, 3.33500685196048792795422182619, 3.61632512735925350678203945908, 3.72773915153050512239087083157

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

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