↑

Properties

Label 24-75e12-1.1-c8e12-0-1
Degree $24$
Conductor $3.168\times 10^{22}$
Sign $1$
Analytic cond. $6.61780\times 10^{17}$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.89e4·11-s + 2.67e4·16-s + 5.65e6·31-s + 1.79e6·41-s + 3.36e7·61-s + 3.35e8·71-s − 1.43e7·81-s + 4.21e8·101-s + 2.41e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2.38e9·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 6.07·11-s + 0.408·16-s + 6.12·31-s + 0.636·41-s + 2.43·61-s + 13.2·71-s − 1/3·81-s + 4.05·101-s + 11.2·121-s − 2.48·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(6.61780\times 10^{17}\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{24} ,\ ( \ : [4]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(51.01951714\)
\(L(\frac12)\) \(\approx\) \(51.01951714\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{14} T^{4} )^{3} \)
5 \( 1 \)
good2 \( 1 - 26775 T^{4} - 10647063 p^{7} T^{8} - 23018182999 p^{12} T^{12} - 10647063 p^{39} T^{16} - 26775 p^{64} T^{20} + p^{96} T^{24} \)
7 \( 1 + 42175424104998 T^{4} + \)\(65\!\cdots\!99\)\( p^{4} T^{8} + \)\(62\!\cdots\!04\)\( p^{8} T^{12} + \)\(65\!\cdots\!99\)\( p^{36} T^{16} + 42175424104998 p^{64} T^{20} + p^{96} T^{24} \)
11 \( ( 1 + 22230 T + 631965711 T^{2} + 9044676988596 T^{3} + 631965711 p^{8} T^{4} + 22230 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
13 \( 1 + 898682346818603430 T^{4} + \)\(86\!\cdots\!51\)\( T^{8} + \)\(75\!\cdots\!16\)\( T^{12} + \)\(86\!\cdots\!51\)\( p^{32} T^{16} + 898682346818603430 p^{64} T^{20} + p^{96} T^{24} \)
17 \( 1 + 28553320315406313414 T^{4} + \)\(38\!\cdots\!15\)\( T^{8} + \)\(23\!\cdots\!80\)\( T^{12} + \)\(38\!\cdots\!15\)\( p^{32} T^{16} + 28553320315406313414 p^{64} T^{20} + p^{96} T^{24} \)
19 \( ( 1 - 1765943838 p T^{2} + \)\(89\!\cdots\!79\)\( T^{4} - \)\(17\!\cdots\!56\)\( T^{6} + \)\(89\!\cdots\!79\)\( p^{16} T^{8} - 1765943838 p^{33} T^{10} + p^{48} T^{12} )^{2} \)
23 \( 1 - \)\(14\!\cdots\!22\)\( T^{4} + \)\(13\!\cdots\!79\)\( T^{8} - \)\(84\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!79\)\( p^{32} T^{16} - \)\(14\!\cdots\!22\)\( p^{64} T^{20} + p^{96} T^{24} \)
29 \( ( 1 - 339394613190 T^{2} + \)\(63\!\cdots\!71\)\( T^{4} - \)\(17\!\cdots\!44\)\( T^{6} + \)\(63\!\cdots\!71\)\( p^{16} T^{8} - 339394613190 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
31 \( ( 1 - 1414674 T + 2627173640367 T^{2} - 2358486313511583868 T^{3} + 2627173640367 p^{8} T^{4} - 1414674 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
37 \( 1 + \)\(50\!\cdots\!18\)\( T^{4} + \)\(39\!\cdots\!19\)\( T^{8} + \)\(13\!\cdots\!84\)\( T^{12} + \)\(39\!\cdots\!19\)\( p^{32} T^{16} + \)\(50\!\cdots\!18\)\( p^{64} T^{20} + p^{96} T^{24} \)
41 \( ( 1 - 449514 T + 11383422179727 T^{2} + 6249272808988527732 T^{3} + 11383422179727 p^{8} T^{4} - 449514 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
43 \( 1 + \)\(47\!\cdots\!94\)\( T^{4} - \)\(91\!\cdots\!85\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{12} - \)\(91\!\cdots\!85\)\( p^{32} T^{16} + \)\(47\!\cdots\!94\)\( p^{64} T^{20} + p^{96} T^{24} \)
47 \( 1 - \)\(10\!\cdots\!26\)\( T^{4} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(57\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!15\)\( p^{32} T^{16} - \)\(10\!\cdots\!26\)\( p^{64} T^{20} + p^{96} T^{24} \)
53 \( 1 - \)\(53\!\cdots\!42\)\( T^{4} - \)\(15\!\cdots\!41\)\( T^{8} - \)\(54\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!41\)\( p^{32} T^{16} - \)\(53\!\cdots\!42\)\( p^{64} T^{20} + p^{96} T^{24} \)
59 \( ( 1 - 8937071780718 p T^{2} + \)\(24\!\cdots\!21\)\( p T^{4} - \)\(25\!\cdots\!76\)\( T^{6} + \)\(24\!\cdots\!21\)\( p^{17} T^{8} - 8937071780718 p^{33} T^{10} + p^{48} T^{12} )^{2} \)
61 \( ( 1 - 8412462 T + 545868881684559 T^{2} - \)\(30\!\cdots\!36\)\( T^{3} + 545868881684559 p^{8} T^{4} - 8412462 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
67 \( 1 - \)\(29\!\cdots\!86\)\( T^{4} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!15\)\( p^{32} T^{16} - \)\(29\!\cdots\!86\)\( p^{64} T^{20} + p^{96} T^{24} \)
71 \( ( 1 - 83876616 T + 4073510682940035 T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + 4073510682940035 p^{8} T^{4} - 83876616 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
73 \( 1 - \)\(16\!\cdots\!22\)\( T^{4} + \)\(17\!\cdots\!79\)\( T^{8} - \)\(12\!\cdots\!76\)\( T^{12} + \)\(17\!\cdots\!79\)\( p^{32} T^{16} - \)\(16\!\cdots\!22\)\( p^{64} T^{20} + p^{96} T^{24} \)
79 \( ( 1 - 569456470757466 T^{2} + \)\(53\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!15\)\( p^{16} T^{8} - 569456470757466 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
83 \( 1 + \)\(37\!\cdots\!10\)\( T^{4} + \)\(56\!\cdots\!11\)\( T^{8} - \)\(11\!\cdots\!84\)\( T^{12} + \)\(56\!\cdots\!11\)\( p^{32} T^{16} + \)\(37\!\cdots\!10\)\( p^{64} T^{20} + p^{96} T^{24} \)
89 \( ( 1 - 4020448845717402 T^{2} + \)\(41\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(41\!\cdots\!99\)\( p^{16} T^{8} - 4020448845717402 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
97 \( 1 + \)\(45\!\cdots\!74\)\( T^{4} - \)\(27\!\cdots\!85\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{12} - \)\(27\!\cdots\!85\)\( p^{32} T^{16} + \)\(45\!\cdots\!74\)\( p^{64} T^{20} + p^{96} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.50042337654990188534691797954, −3.45314645987716467138904074796, −3.41088891359944431159373617237, −3.00795172328509703705906214110, −2.74398035088432893174256551724, −2.69908097165599896313928700153, −2.62661248531638457377039495823, −2.56358985999048050156391030811, −2.48604894736160365742898166555, −2.45047188648397308908985977396, −2.43878956090764404474130414024, −2.18369374363524333037576234560, −1.89121083728310959409013135052, −1.79555374918345209586956853535, −1.75667732096177841787111949032, −1.34151900786503870083457037207, −1.30180618150367930940723740833, −0.913155551037018025213283511916, −0.833474641384450957150919284914, −0.71104220403058636292589678083, −0.64049231231612469305516137913, −0.51019784862200433906897018539, −0.44474572130539680780612715354, −0.35893166316332525648610055678, −0.35435901700974568027700445281, 0.35435901700974568027700445281, 0.35893166316332525648610055678, 0.44474572130539680780612715354, 0.51019784862200433906897018539, 0.64049231231612469305516137913, 0.71104220403058636292589678083, 0.833474641384450957150919284914, 0.913155551037018025213283511916, 1.30180618150367930940723740833, 1.34151900786503870083457037207, 1.75667732096177841787111949032, 1.79555374918345209586956853535, 1.89121083728310959409013135052, 2.18369374363524333037576234560, 2.43878956090764404474130414024, 2.45047188648397308908985977396, 2.48604894736160365742898166555, 2.56358985999048050156391030811, 2.62661248531638457377039495823, 2.69908097165599896313928700153, 2.74398035088432893174256551724, 3.00795172328509703705906214110, 3.41088891359944431159373617237, 3.45314645987716467138904074796, 3.50042337654990188534691797954

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

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