Properties

Label 24-475e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.319\times 10^{32}$
Sign $1$
Analytic cond. $8.86438\times 10^{6}$
Root an. cond. $1.94753$
Motivic weight $1$
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  − 3·9-s + 24·19-s + 6·29-s + 18·31-s + 42·41-s − 36·49-s − 24·59-s − 24·61-s + 2·64-s − 12·71-s + 78·79-s + 30·81-s + 24·89-s − 42·101-s − 36·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s − 72·171-s + ⋯
L(s)  = 1  − 9-s + 5.50·19-s + 1.11·29-s + 3.23·31-s + 6.55·41-s − 5.14·49-s − 3.12·59-s − 3.07·61-s + 1/4·64-s − 1.42·71-s + 8.77·79-s + 10/3·81-s + 2.54·89-s − 4.17·101-s − 3.44·109-s + 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3·169-s − 5.50·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.86438\times 10^{6}\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{24} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.386236164\)
\(L(\frac12)\) \(\approx\) \(2.386236164\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( ( 1 - 12 T + 78 T^{2} - 385 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good2 \( 1 - p T^{6} - 27 T^{10} + 21 T^{12} - 27 p^{2} T^{14} - p^{7} T^{18} + p^{12} T^{24} \)
3 \( 1 + p T^{2} - 7 p T^{4} - 37 T^{6} + 106 p T^{8} + 68 p T^{10} - 3095 T^{12} + 68 p^{3} T^{14} + 106 p^{5} T^{16} - 37 p^{6} T^{18} - 7 p^{9} T^{20} + p^{11} T^{22} + p^{12} T^{24} \)
7 \( 1 + 36 T^{2} + 720 T^{4} + 10294 T^{6} + 115596 T^{8} + 1061604 T^{10} + 8125419 T^{12} + 1061604 p^{2} T^{14} + 115596 p^{4} T^{16} + 10294 p^{6} T^{18} + 720 p^{8} T^{20} + 36 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 216 p T^{7} + 312 p^{2} T^{8} - 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( 1 - 3 p T^{2} + 1050 T^{4} - 22688 T^{6} + 407205 T^{8} - 6429069 T^{10} + 88728201 T^{12} - 6429069 p^{2} T^{14} + 407205 p^{4} T^{16} - 22688 p^{6} T^{18} + 1050 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
17 \( 1 + 9 T^{2} - 666 T^{4} - 5204 T^{6} + 259713 T^{8} + 947187 T^{10} - 73705119 T^{12} + 947187 p^{2} T^{14} + 259713 p^{4} T^{16} - 5204 p^{6} T^{18} - 666 p^{8} T^{20} + 9 p^{10} T^{22} + p^{12} T^{24} \)
23 \( 1 - 36 T^{2} + 1800 T^{4} - 54686 T^{6} + 1497096 T^{8} - 40942800 T^{10} + 850507203 T^{12} - 40942800 p^{2} T^{14} + 1497096 p^{4} T^{16} - 54686 p^{6} T^{18} + 1800 p^{8} T^{20} - 36 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 3 T + 36 T^{2} - 378 T^{3} + 1872 T^{4} - 9921 T^{5} + 94159 T^{6} - 9921 p T^{7} + 1872 p^{2} T^{8} - 378 p^{3} T^{9} + 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 3402 p T^{7} + 2187 p^{2} T^{8} + 119 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 180 T^{2} + 14760 T^{4} - 700417 T^{6} + 14760 p^{2} T^{8} - 180 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 28401 p T^{7} - 4707 p^{2} T^{8} - 180 p^{3} T^{9} + 162 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 - 111 T^{2} + 8934 T^{4} - 600344 T^{6} + 34082793 T^{8} - 1785671613 T^{10} + 82357476561 T^{12} - 1785671613 p^{2} T^{14} + 34082793 p^{4} T^{16} - 600344 p^{6} T^{18} + 8934 p^{8} T^{20} - 111 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 - 99 T^{2} + 10350 T^{4} - 635852 T^{6} + 39935457 T^{8} - 1842071193 T^{10} + 97644607185 T^{12} - 1842071193 p^{2} T^{14} + 39935457 p^{4} T^{16} - 635852 p^{6} T^{18} + 10350 p^{8} T^{20} - 99 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 9 T^{2} - 4446 T^{4} - 53048 T^{6} + 294273 p T^{8} + 201951603 T^{10} - 47390910927 T^{12} + 201951603 p^{2} T^{14} + 294273 p^{5} T^{16} - 53048 p^{6} T^{18} - 4446 p^{8} T^{20} + 9 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 12 T + 18 T^{2} - 1080 T^{3} - 7614 T^{4} + 29982 T^{5} + 754273 T^{6} + 29982 p T^{7} - 7614 p^{2} T^{8} - 1080 p^{3} T^{9} + 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 12 T + 24 T^{2} - 586 T^{3} - 4140 T^{4} + 44676 T^{5} + 736335 T^{6} + 44676 p T^{7} - 4140 p^{2} T^{8} - 586 p^{3} T^{9} + 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 204 T^{2} + 28968 T^{4} + 3110554 T^{6} + 279327816 T^{8} + 21830188320 T^{10} + 1553708853819 T^{12} + 21830188320 p^{2} T^{14} + 279327816 p^{4} T^{16} + 3110554 p^{6} T^{18} + 28968 p^{8} T^{20} + 204 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 + 6 T - 36 T^{2} + 594 T^{3} + 3240 T^{4} + 14892 T^{5} + 665785 T^{6} + 14892 p T^{7} + 3240 p^{2} T^{8} + 594 p^{3} T^{9} - 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 - 48 T^{2} + 12480 T^{4} - 723950 T^{6} + 46100448 T^{8} - 5607700848 T^{10} + 86537829843 T^{12} - 5607700848 p^{2} T^{14} + 46100448 p^{4} T^{16} - 723950 p^{6} T^{18} + 12480 p^{8} T^{20} - 48 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 39 T + 708 T^{2} - 8198 T^{3} + 67068 T^{4} - 403623 T^{5} + 2596617 T^{6} - 403623 p T^{7} + 67068 p^{2} T^{8} - 8198 p^{3} T^{9} + 708 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 + 120 T^{2} + 840 T^{4} - 1433786 T^{6} - 93200400 T^{8} + 4028922000 T^{10} + 1046184516987 T^{12} + 4028922000 p^{2} T^{14} - 93200400 p^{4} T^{16} - 1433786 p^{6} T^{18} + 840 p^{8} T^{20} + 120 p^{10} T^{22} + p^{12} T^{24} \)
89 \( ( 1 - 12 T + 54 T^{2} - 1035 T^{3} - 279 T^{4} + 69891 T^{5} + 3961 T^{6} + 69891 p T^{7} - 279 p^{2} T^{8} - 1035 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 - 144 T^{2} + 14004 T^{4} - 1512866 T^{6} + 271998324 T^{8} - 22595563332 T^{10} + 1991132645331 T^{12} - 22595563332 p^{2} T^{14} + 271998324 p^{4} T^{16} - 1512866 p^{6} T^{18} + 14004 p^{8} T^{20} - 144 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.52528006585011357431569521756, −3.49726414552924320090235036788, −3.49693285505752815392716043980, −3.29209889125349117477678312337, −3.28762810398895739182528470043, −3.04610692707993707718089227414, −3.02149094194719624743355376585, −2.94922720973595386765095339685, −2.92758753760774715756995267708, −2.62319164959072620839405836626, −2.56457895591992553472076571191, −2.55268257614048851672369084977, −2.30572224359858359851253845214, −2.28605661249247930497770021977, −2.24772087479509333955522092844, −2.01829419183520630756638039468, −1.64357799668184896537781833160, −1.55176973309267922264124066083, −1.31931840680458621686843238919, −1.25478569462996028645775674884, −1.12496667491855406498180101074, −0.966744213902868536118628487915, −0.896352416052807061168940549451, −0.74981783162543557737218332077, −0.15651967696794221254432718808, 0.15651967696794221254432718808, 0.74981783162543557737218332077, 0.896352416052807061168940549451, 0.966744213902868536118628487915, 1.12496667491855406498180101074, 1.25478569462996028645775674884, 1.31931840680458621686843238919, 1.55176973309267922264124066083, 1.64357799668184896537781833160, 2.01829419183520630756638039468, 2.24772087479509333955522092844, 2.28605661249247930497770021977, 2.30572224359858359851253845214, 2.55268257614048851672369084977, 2.56457895591992553472076571191, 2.62319164959072620839405836626, 2.92758753760774715756995267708, 2.94922720973595386765095339685, 3.02149094194719624743355376585, 3.04610692707993707718089227414, 3.28762810398895739182528470043, 3.29209889125349117477678312337, 3.49693285505752815392716043980, 3.49726414552924320090235036788, 3.52528006585011357431569521756

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

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