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Properties

Label 24-212e12-1.1-c0e12-0-0
Degree $24$
Conductor $8.242\times 10^{27}$
Sign $1$
Analytic cond. $1.96750\times 10^{-12}$
Root an. cond. $0.325271$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s − 2·29-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s − 53-s + 2·58-s − 2·61-s + 4·65-s − 2·73-s + 2·74-s + 2·82-s + 4·85-s + 11·89-s − 2·90-s + 11·97-s + ⋯
L(s)  = 1  − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s − 2·29-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s − 53-s + 2·58-s − 2·61-s + 4·65-s − 2·73-s + 2·74-s + 2·82-s + 4·85-s + 11·89-s − 2·90-s + 11·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 53^{12}\)
Sign: $1$
Analytic conductor: \(1.96750\times 10^{-12}\)
Root analytic conductor: \(0.325271\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 53^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005654542859\)
\(L(\frac12)\) \(\approx\) \(0.005654542859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
23 \( ( 1 - T )^{12}( 1 + T )^{12} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
83 \( ( 1 - T )^{12}( 1 + T )^{12} \)
89 \( ( 1 - T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
97 \( ( 1 - T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
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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

−4.64994119277875929901635562035, −4.64044823959314145601289150623, −4.55754527448973107417259331083, −4.32408372673008109505161212652, −4.31464191777797669294918498492, −4.10586711342160607150286589220, −3.67519076802748006537406798326, −3.63503655773572921655043271559, −3.63366735166351168950377892547, −3.61689346423951193955689431527, −3.51023135819205807345022044099, −3.49383268487478417346412651070, −3.46398464931119889096579178762, −3.18194873043837120386837667682, −2.86396049246181112621873156414, −2.85087819006337346933044994986, −2.59524502828890696442154544351, −2.53152133158119985577809829010, −2.17974758046994104071658256711, −2.17667313944176465959251688953, −2.09646339975387941301758385634, −1.99952962217976455083950035634, −1.79737769210714574277173780015, −1.39374763737186439145944168255, −1.22590541841747273789943295818, 1.22590541841747273789943295818, 1.39374763737186439145944168255, 1.79737769210714574277173780015, 1.99952962217976455083950035634, 2.09646339975387941301758385634, 2.17667313944176465959251688953, 2.17974758046994104071658256711, 2.53152133158119985577809829010, 2.59524502828890696442154544351, 2.85087819006337346933044994986, 2.86396049246181112621873156414, 3.18194873043837120386837667682, 3.46398464931119889096579178762, 3.49383268487478417346412651070, 3.51023135819205807345022044099, 3.61689346423951193955689431527, 3.63366735166351168950377892547, 3.63503655773572921655043271559, 3.67519076802748006537406798326, 4.10586711342160607150286589220, 4.31464191777797669294918498492, 4.32408372673008109505161212652, 4.55754527448973107417259331083, 4.64044823959314145601289150623, 4.64994119277875929901635562035

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

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