Properties

Label 24-475e12-1.1-c1e12-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·7-s − 16·11-s + 13·16-s − 20·17-s + 12·23-s − 4·43-s + 44·47-s + 72·49-s − 24·61-s − 28·73-s − 192·77-s + 30·81-s + 36·83-s − 24·101-s + 156·112-s − 240·119-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + 167-s + ⋯
L(s)  = 1  + 4.53·7-s − 4.82·11-s + 13/4·16-s − 4.85·17-s + 2.50·23-s − 0.609·43-s + 6.41·47-s + 72/7·49-s − 3.07·61-s − 3.27·73-s − 21.8·77-s + 10/3·81-s + 3.95·83-s − 2.38·101-s + 14.7·112-s − 22.0·119-s + 4·121-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 + 11.3·161-s + 0.0783·163-s + 0.0773·167-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

\(L(1)\) \(\approx\) \(3.352732062\)
\(L(\frac12)\) \(\approx\) \(3.352732062\)
\(L(\frac{3}{2})\) 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 \( 1 - 70 T^{2} + 2503 T^{4} - 58228 T^{6} + 2503 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( 1 - 13 T^{4} + 95 T^{8} - 447 T^{12} + 95 p^{4} T^{16} - 13 p^{8} T^{20} + p^{12} T^{24} \)
3 \( 1 - 10 p T^{4} + 527 T^{8} - 5684 T^{12} + 527 p^{4} T^{16} - 10 p^{9} T^{20} + p^{12} T^{24} \)
7 \( ( 1 - 6 T + 18 T^{2} - 50 T^{3} + 95 T^{4} - 124 T^{5} + 284 T^{6} - 124 p T^{7} + 95 p^{2} T^{8} - 50 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( ( 1 + 4 T + 29 T^{2} + 68 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
13 \( 1 + 370 T^{4} + 115567 T^{8} + 20399596 T^{12} + 115567 p^{4} T^{16} + 370 p^{8} T^{20} + p^{12} T^{24} \)
17 \( ( 1 - 2 T - 9 T^{2} + 132 T^{3} - 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 12 T + 83 T^{2} + 392 T^{3} + 83 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( ( 1 - 6 T + 18 T^{2} - 146 T^{3} + 1407 T^{4} - 4732 T^{5} + 13724 T^{6} - 4732 p T^{7} + 1407 p^{2} T^{8} - 146 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
29 \( ( 1 + 42 T^{2} + 1751 T^{4} + 66092 T^{6} + 1751 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 38 T^{2} - 209 T^{4} + 40012 T^{6} - 209 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( 1 + 3874 T^{4} + 8340271 T^{8} + 13273985356 T^{12} + 8340271 p^{4} T^{16} + 3874 p^{8} T^{20} + p^{12} T^{24} \)
41 \( ( 1 - 70 T^{2} + 2783 T^{4} - 76308 T^{6} + 2783 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 2 T + 2 T^{2} - 450 T^{3} - 1705 T^{4} + 9508 T^{5} + 123676 T^{6} + 9508 p T^{7} - 1705 p^{2} T^{8} - 450 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 - 22 T + 242 T^{2} - 2130 T^{3} + 15215 T^{4} - 89948 T^{5} + 565276 T^{6} - 89948 p T^{7} + 15215 p^{2} T^{8} - 2130 p^{3} T^{9} + 242 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
53 \( 1 - 190 T^{4} - 3502993 T^{8} + 27305462796 T^{12} - 3502993 p^{4} T^{16} - 190 p^{8} T^{20} + p^{12} T^{24} \)
59 \( ( 1 + 178 T^{2} + 17111 T^{4} + 1136220 T^{6} + 17111 p^{2} T^{8} + 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 6 T + 95 T^{2} + 272 T^{3} + 95 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( 1 - 7918 T^{4} + 3136175 T^{8} + 119483761068 T^{12} + 3136175 p^{4} T^{16} - 7918 p^{8} T^{20} + p^{12} T^{24} \)
71 \( ( 1 - 286 T^{2} + 38687 T^{4} - 3312324 T^{6} + 38687 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 14 T + 98 T^{2} + 654 T^{3} - 4753 T^{4} - 105452 T^{5} - 796676 T^{6} - 105452 p T^{7} - 4753 p^{2} T^{8} + 654 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 274 T^{2} + 37103 T^{4} + 3373884 T^{6} + 37103 p^{2} T^{8} + 274 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 18 T + 162 T^{2} - 1646 T^{3} + 18711 T^{4} - 175860 T^{5} + 1488956 T^{6} - 175860 p T^{7} + 18711 p^{2} T^{8} - 1646 p^{3} T^{9} + 162 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 186 T^{2} + 15695 T^{4} + 917036 T^{6} + 15695 p^{2} T^{8} + 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( 1 - 2382 T^{4} - 45699681 T^{8} + 56230677484 T^{12} - 45699681 p^{4} T^{16} - 2382 p^{8} T^{20} + p^{12} T^{24} \)
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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

−3.78229744774062157215098114197, −3.48622837549928553892867199681, −3.44051367281496132874004090384, −3.26652989929715810538784665368, −3.23188006333097075240491166034, −3.01888638299714304047381301464, −3.00483126178853947453695930712, −2.83258955298845706409733335554, −2.68620898625272274669700106703, −2.50166692938833560718330483142, −2.45198009462632361819358735052, −2.41538800589178504183820695628, −2.40828679500271495065038941645, −2.24567092219684246859713393160, −2.18101855294755992936398226964, −1.96928119623476520534110042526, −1.88420899210643139109019360402, −1.53913190337415048040514718727, −1.51253193339586825100045892686, −1.33982368712425331839247688337, −1.22664173459353405325987202266, −1.08027774955413751711265895587, −0.796660393342935336524064328510, −0.54264711455239919316528614581, −0.20801180096831809538978631265, 0.20801180096831809538978631265, 0.54264711455239919316528614581, 0.796660393342935336524064328510, 1.08027774955413751711265895587, 1.22664173459353405325987202266, 1.33982368712425331839247688337, 1.51253193339586825100045892686, 1.53913190337415048040514718727, 1.88420899210643139109019360402, 1.96928119623476520534110042526, 2.18101855294755992936398226964, 2.24567092219684246859713393160, 2.40828679500271495065038941645, 2.41538800589178504183820695628, 2.45198009462632361819358735052, 2.50166692938833560718330483142, 2.68620898625272274669700106703, 2.83258955298845706409733335554, 3.00483126178853947453695930712, 3.01888638299714304047381301464, 3.23188006333097075240491166034, 3.26652989929715810538784665368, 3.44051367281496132874004090384, 3.48622837549928553892867199681, 3.78229744774062157215098114197

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

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