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Properties

Label 24-2268e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.852\times 10^{40}$
Sign $1$
Analytic cond. $1.24463\times 10^{15}$
Root an. cond. $4.25559$
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  − 18·19-s − 30·25-s − 24·37-s + 6·43-s − 18·61-s + 54·73-s + 24·79-s − 36·97-s − 12·109-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4.12·19-s − 6·25-s − 3.94·37-s + 0.914·43-s − 2.30·61-s + 6.32·73-s + 2.70·79-s − 3.65·97-s − 1.14·109-s + 3.81·121-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 − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{48} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.24463\times 10^{15}\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{48} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01200314284\)
\(L(\frac12)\) \(\approx\) \(0.01200314284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 - p T^{3} + p^{3} T^{6} )^{2} \)
good5 \( ( 1 + 3 p T^{2} + 129 T^{4} + 763 T^{6} + 129 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 21 T^{2} + 321 T^{4} - 3913 T^{6} + 321 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 + 18 T^{2} + 90 T^{4} + 1134 T^{5} + 1141 T^{6} + 1134 p T^{7} + 90 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 12 T^{2} - 624 T^{4} - 4882 T^{6} + 283284 T^{8} + 1089036 T^{10} - 84522621 T^{12} + 1089036 p^{2} T^{14} + 283284 p^{4} T^{16} - 4882 p^{6} T^{18} - 624 p^{8} T^{20} + 12 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 9 T + 72 T^{2} + 405 T^{3} + 117 p T^{4} + 12546 T^{5} + 56077 T^{6} + 12546 p T^{7} + 117 p^{3} T^{8} + 405 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 84 T^{2} + 3372 T^{4} - 89845 T^{6} + 3372 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( 1 + 21 T^{2} - 2040 T^{4} - 16205 T^{6} + 3351627 T^{8} + 11952612 T^{10} - 3132495471 T^{12} + 11952612 p^{2} T^{14} + 3351627 p^{4} T^{16} - 16205 p^{6} T^{18} - 2040 p^{8} T^{20} + 21 p^{10} T^{22} + p^{12} T^{24} \)
31 \( ( 1 + 30 T^{2} - 30 T^{4} - 1890 T^{5} - 30359 T^{6} - 1890 p T^{7} - 30 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 12 T + 48 T^{2} + 130 T^{3} - 468 T^{4} - 19764 T^{5} - 175701 T^{6} - 19764 p T^{7} - 468 p^{2} T^{8} + 130 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 60 T^{2} - 1236 T^{4} + 102974 T^{6} + 3463560 T^{8} - 144625752 T^{10} - 1347317805 T^{12} - 144625752 p^{2} T^{14} + 3463560 p^{4} T^{16} + 102974 p^{6} T^{18} - 1236 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 - 3 T - 102 T^{2} + 157 T^{3} + 6813 T^{4} - 4104 T^{5} - 327405 T^{6} - 4104 p T^{7} + 6813 p^{2} T^{8} + 157 p^{3} T^{9} - 102 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 - 267 T^{2} + 40920 T^{4} - 4343125 T^{6} + 352510407 T^{8} - 22655106144 T^{10} + 1180938336801 T^{12} - 22655106144 p^{2} T^{14} + 352510407 p^{4} T^{16} - 4343125 p^{6} T^{18} + 40920 p^{8} T^{20} - 267 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 84 T^{2} + 2136 T^{4} - 35042 T^{6} - 8443404 T^{8} - 556134516 T^{10} - 29094135357 T^{12} - 556134516 p^{2} T^{14} - 8443404 p^{4} T^{16} - 35042 p^{6} T^{18} + 2136 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \)
59 \( 1 - 51 T^{2} - 1380 T^{4} + 158351 T^{6} - 7224741 T^{8} + 76185540 T^{10} + 30324904161 T^{12} + 76185540 p^{2} T^{14} - 7224741 p^{4} T^{16} + 158351 p^{6} T^{18} - 1380 p^{8} T^{20} - 51 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 + 9 T + 135 T^{2} + 972 T^{3} + 9153 T^{4} + 113283 T^{5} + 772702 T^{6} + 113283 p T^{7} + 9153 p^{2} T^{8} + 972 p^{3} T^{9} + 135 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 54 T^{2} - 994 T^{3} - 702 T^{4} + 26838 T^{5} + 653163 T^{6} + 26838 p T^{7} - 702 p^{2} T^{8} - 994 p^{3} T^{9} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 237 T^{2} + 29877 T^{4} - 2533201 T^{6} + 29877 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 27 T + 522 T^{2} - 7533 T^{3} + 94005 T^{4} - 988578 T^{5} + 9101239 T^{6} - 988578 p T^{7} + 94005 p^{2} T^{8} - 7533 p^{3} T^{9} + 522 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 12 T - 78 T^{2} + 374 T^{3} + 13518 T^{4} + 8802 T^{5} - 1498701 T^{6} + 8802 p T^{7} + 13518 p^{2} T^{8} + 374 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 123 T^{2} + 2544 T^{4} + 684275 T^{6} - 65597985 T^{8} + 1444043232 T^{10} + 90501353529 T^{12} + 1444043232 p^{2} T^{14} - 65597985 p^{4} T^{16} + 684275 p^{6} T^{18} + 2544 p^{8} T^{20} - 123 p^{10} T^{22} + p^{12} T^{24} \)
89 \( ( 1 - 151 T^{2} + 14880 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} )^{3} \)
97 \( ( 1 + 18 T + 351 T^{2} + 4374 T^{3} + 56862 T^{4} + 696690 T^{5} + 7120267 T^{6} + 696690 p T^{7} + 56862 p^{2} T^{8} + 4374 p^{3} T^{9} + 351 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−2.78449525256089197675161579595, −2.74764003921527159476758773810, −2.53426768887662093860715880930, −2.36004418525042068496884374316, −2.28068363061862505486325982683, −2.14870747559731622628322372845, −2.07599065288715072640659984105, −2.06601525590024927363583850007, −2.06572009090414749623087375976, −2.03949752643348484423617217031, −1.99204817546706812200669077018, −1.87023043021037423756629707365, −1.82332575867063722240861971157, −1.65744477322027963415654153080, −1.59892729031463957464425515418, −1.24675583782651520516743175830, −1.24474648939276879005306694335, −1.07476081924116941288481040071, −0.956247728042987914978443560635, −0.893089989831959847294760527975, −0.790839225448033734021718200524, −0.44006131761462419218757905423, −0.22006510740895317834105932372, −0.13986280500610151876192488249, −0.02907096130640238316471021952, 0.02907096130640238316471021952, 0.13986280500610151876192488249, 0.22006510740895317834105932372, 0.44006131761462419218757905423, 0.790839225448033734021718200524, 0.893089989831959847294760527975, 0.956247728042987914978443560635, 1.07476081924116941288481040071, 1.24474648939276879005306694335, 1.24675583782651520516743175830, 1.59892729031463957464425515418, 1.65744477322027963415654153080, 1.82332575867063722240861971157, 1.87023043021037423756629707365, 1.99204817546706812200669077018, 2.03949752643348484423617217031, 2.06572009090414749623087375976, 2.06601525590024927363583850007, 2.07599065288715072640659984105, 2.14870747559731622628322372845, 2.28068363061862505486325982683, 2.36004418525042068496884374316, 2.53426768887662093860715880930, 2.74764003921527159476758773810, 2.78449525256089197675161579595

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

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