Properties

Degree 24
Conductor $ 2^{12} \cdot 5^{12} \cdot 11^{12} \cdot 43^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 12·2-s + 3·3-s + 78·4-s + 12·5-s + 36·6-s + 8·7-s + 364·8-s − 9-s + 144·10-s − 12·11-s + 234·12-s + 16·13-s + 96·14-s + 36·15-s + 1.36e3·16-s + 18·17-s − 12·18-s − 4·19-s + 936·20-s + 24·21-s − 144·22-s + 8·23-s + 1.09e3·24-s + 78·25-s + 192·26-s − 13·27-s + 624·28-s + ⋯
L(s)  = 1  + 8.48·2-s + 1.73·3-s + 39·4-s + 5.36·5-s + 14.6·6-s + 3.02·7-s + 128.·8-s − 1/3·9-s + 45.5·10-s − 3.61·11-s + 67.5·12-s + 4.43·13-s + 25.6·14-s + 9.29·15-s + 341.·16-s + 4.36·17-s − 2.82·18-s − 0.917·19-s + 209.·20-s + 5.23·21-s − 30.7·22-s + 1.66·23-s + 222.·24-s + 78/5·25-s + 37.6·26-s − 2.50·27-s + 117.·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 11^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 11^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{12} \cdot 5^{12} \cdot 11^{12} \cdot 43^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4730} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{12} \cdot 5^{12} \cdot 11^{12} \cdot 43^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $4.833848046e6$
$L(\frac12)$  $\approx$  $4.833848046e6$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( ( 1 - T )^{12} \)
5 \( ( 1 - T )^{12} \)
11 \( ( 1 + T )^{12} \)
43 \( ( 1 - T )^{12} \)
good3 \( 1 - p T + 10 T^{2} - 20 T^{3} + 61 T^{4} - 118 T^{5} + 35 p^{2} T^{6} - 574 T^{7} + 1319 T^{8} - 2134 T^{9} + 520 p^{2} T^{10} - 7469 T^{11} + 15260 T^{12} - 7469 p T^{13} + 520 p^{4} T^{14} - 2134 p^{3} T^{15} + 1319 p^{4} T^{16} - 574 p^{5} T^{17} + 35 p^{8} T^{18} - 118 p^{7} T^{19} + 61 p^{8} T^{20} - 20 p^{9} T^{21} + 10 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \)
7 \( 1 - 8 T + 48 T^{2} - 195 T^{3} + 734 T^{4} - 2357 T^{5} + 7837 T^{6} - 23232 T^{7} + 70531 T^{8} - 189801 T^{9} + 531413 T^{10} - 1351363 T^{11} + 531680 p T^{12} - 1351363 p T^{13} + 531413 p^{2} T^{14} - 189801 p^{3} T^{15} + 70531 p^{4} T^{16} - 23232 p^{5} T^{17} + 7837 p^{6} T^{18} - 2357 p^{7} T^{19} + 734 p^{8} T^{20} - 195 p^{9} T^{21} + 48 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 16 T + 190 T^{2} - 1610 T^{3} + 11714 T^{4} - 72126 T^{5} + 404502 T^{6} - 2041308 T^{7} + 9651151 T^{8} - 42172772 T^{9} + 174989468 T^{10} - 679932320 T^{11} + 2527194876 T^{12} - 679932320 p T^{13} + 174989468 p^{2} T^{14} - 42172772 p^{3} T^{15} + 9651151 p^{4} T^{16} - 2041308 p^{5} T^{17} + 404502 p^{6} T^{18} - 72126 p^{7} T^{19} + 11714 p^{8} T^{20} - 1610 p^{9} T^{21} + 190 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 18 T + 234 T^{2} - 137 p T^{3} + 19874 T^{4} - 147979 T^{5} + 992157 T^{6} - 6051758 T^{7} + 34033979 T^{8} - 177362485 T^{9} + 861964541 T^{10} - 3916766233 T^{11} + 16680080260 T^{12} - 3916766233 p T^{13} + 861964541 p^{2} T^{14} - 177362485 p^{3} T^{15} + 34033979 p^{4} T^{16} - 6051758 p^{5} T^{17} + 992157 p^{6} T^{18} - 147979 p^{7} T^{19} + 19874 p^{8} T^{20} - 137 p^{10} T^{21} + 234 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 4 T + 88 T^{2} + 199 T^{3} + 3814 T^{4} + 7351 T^{5} + 133259 T^{6} + 250836 T^{7} + 3633077 T^{8} + 6088153 T^{9} + 83260779 T^{10} + 140182317 T^{11} + 1720895564 T^{12} + 140182317 p T^{13} + 83260779 p^{2} T^{14} + 6088153 p^{3} T^{15} + 3633077 p^{4} T^{16} + 250836 p^{5} T^{17} + 133259 p^{6} T^{18} + 7351 p^{7} T^{19} + 3814 p^{8} T^{20} + 199 p^{9} T^{21} + 88 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 8 T + 222 T^{2} - 1622 T^{3} + 23758 T^{4} - 156190 T^{5} + 1609758 T^{6} - 410252 p T^{7} + 76497695 T^{8} - 397398220 T^{9} + 2677343844 T^{10} - 12246544760 T^{11} + 70659812068 T^{12} - 12246544760 p T^{13} + 2677343844 p^{2} T^{14} - 397398220 p^{3} T^{15} + 76497695 p^{4} T^{16} - 410252 p^{6} T^{17} + 1609758 p^{6} T^{18} - 156190 p^{7} T^{19} + 23758 p^{8} T^{20} - 1622 p^{9} T^{21} + 222 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 20 T + 326 T^{2} - 3606 T^{3} + 34427 T^{4} - 269478 T^{5} + 1908284 T^{6} - 12081496 T^{7} + 72893767 T^{8} - 420033604 T^{9} + 2390489470 T^{10} - 13286095452 T^{11} + 72603739482 T^{12} - 13286095452 p T^{13} + 2390489470 p^{2} T^{14} - 420033604 p^{3} T^{15} + 72893767 p^{4} T^{16} - 12081496 p^{5} T^{17} + 1908284 p^{6} T^{18} - 269478 p^{7} T^{19} + 34427 p^{8} T^{20} - 3606 p^{9} T^{21} + 326 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 5 T + 163 T^{2} - 392 T^{3} + 12720 T^{4} - 10628 T^{5} + 718551 T^{6} + 168341 T^{7} + 33010935 T^{8} + 27583730 T^{9} + 1295215302 T^{10} + 1327436284 T^{11} + 43461407152 T^{12} + 1327436284 p T^{13} + 1295215302 p^{2} T^{14} + 27583730 p^{3} T^{15} + 33010935 p^{4} T^{16} + 168341 p^{5} T^{17} + 718551 p^{6} T^{18} - 10628 p^{7} T^{19} + 12720 p^{8} T^{20} - 392 p^{9} T^{21} + 163 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 19 T + 399 T^{2} - 4942 T^{3} + 62514 T^{4} - 594934 T^{5} + 5734375 T^{6} - 45439323 T^{7} + 366555127 T^{8} - 68640566 p T^{9} + 18066209682 T^{10} - 112976519612 T^{11} + 730091241356 T^{12} - 112976519612 p T^{13} + 18066209682 p^{2} T^{14} - 68640566 p^{4} T^{15} + 366555127 p^{4} T^{16} - 45439323 p^{5} T^{17} + 5734375 p^{6} T^{18} - 594934 p^{7} T^{19} + 62514 p^{8} T^{20} - 4942 p^{9} T^{21} + 399 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 16 T + 228 T^{2} - 2402 T^{3} + 27018 T^{4} - 249114 T^{5} + 2281756 T^{6} - 18226192 T^{7} + 147465407 T^{8} - 1080823472 T^{9} + 7939156288 T^{10} - 52693816004 T^{11} + 351179300780 T^{12} - 52693816004 p T^{13} + 7939156288 p^{2} T^{14} - 1080823472 p^{3} T^{15} + 147465407 p^{4} T^{16} - 18226192 p^{5} T^{17} + 2281756 p^{6} T^{18} - 249114 p^{7} T^{19} + 27018 p^{8} T^{20} - 2402 p^{9} T^{21} + 228 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + T + 114 T^{2} + 244 T^{3} + 9344 T^{4} + 23575 T^{5} + 655493 T^{6} + 1110728 T^{7} + 35897107 T^{8} + 52669935 T^{9} + 1857302769 T^{10} + 1354844891 T^{11} + 90478438248 T^{12} + 1354844891 p T^{13} + 1857302769 p^{2} T^{14} + 52669935 p^{3} T^{15} + 35897107 p^{4} T^{16} + 1110728 p^{5} T^{17} + 655493 p^{6} T^{18} + 23575 p^{7} T^{19} + 9344 p^{8} T^{20} + 244 p^{9} T^{21} + 114 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 11 T + 272 T^{2} - 2622 T^{3} + 43585 T^{4} - 377096 T^{5} + 4842391 T^{6} - 38473126 T^{7} + 419861413 T^{8} - 3062483462 T^{9} + 553437466 p T^{10} - 197228803387 T^{11} + 1705164067976 T^{12} - 197228803387 p T^{13} + 553437466 p^{3} T^{14} - 3062483462 p^{3} T^{15} + 419861413 p^{4} T^{16} - 38473126 p^{5} T^{17} + 4842391 p^{6} T^{18} - 377096 p^{7} T^{19} + 43585 p^{8} T^{20} - 2622 p^{9} T^{21} + 272 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 11 T + 410 T^{2} + 3848 T^{3} + 1370 p T^{4} + 652367 T^{5} + 10202585 T^{6} + 71195372 T^{7} + 936853399 T^{8} + 5751551047 T^{9} + 68575048349 T^{10} + 382685049041 T^{11} + 4302652813284 T^{12} + 382685049041 p T^{13} + 68575048349 p^{2} T^{14} + 5751551047 p^{3} T^{15} + 936853399 p^{4} T^{16} + 71195372 p^{5} T^{17} + 10202585 p^{6} T^{18} + 652367 p^{7} T^{19} + 1370 p^{9} T^{20} + 3848 p^{9} T^{21} + 410 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 18 T + 540 T^{2} - 7190 T^{3} + 127321 T^{4} - 1364142 T^{5} + 18495558 T^{6} - 169865110 T^{7} + 32061979 p T^{8} - 16033931952 T^{9} + 163672721502 T^{10} - 1212570278980 T^{11} + 11092826478254 T^{12} - 1212570278980 p T^{13} + 163672721502 p^{2} T^{14} - 16033931952 p^{3} T^{15} + 32061979 p^{5} T^{16} - 169865110 p^{5} T^{17} + 18495558 p^{6} T^{18} - 1364142 p^{7} T^{19} + 127321 p^{8} T^{20} - 7190 p^{9} T^{21} + 540 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 10 T + 318 T^{2} + 3250 T^{3} + 55350 T^{4} + 454246 T^{5} + 6110150 T^{6} + 37277342 T^{7} + 433674287 T^{8} + 1819168700 T^{9} + 22259029868 T^{10} + 46750312876 T^{11} + 1215427907796 T^{12} + 46750312876 p T^{13} + 22259029868 p^{2} T^{14} + 1819168700 p^{3} T^{15} + 433674287 p^{4} T^{16} + 37277342 p^{5} T^{17} + 6110150 p^{6} T^{18} + 454246 p^{7} T^{19} + 55350 p^{8} T^{20} + 3250 p^{9} T^{21} + 318 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 2 T + 440 T^{2} + 1279 T^{3} + 102850 T^{4} + 331941 T^{5} + 16564343 T^{6} + 53384690 T^{7} + 2020488169 T^{8} + 6192873709 T^{9} + 195311727749 T^{10} + 555125727073 T^{11} + 15326747025720 T^{12} + 555125727073 p T^{13} + 195311727749 p^{2} T^{14} + 6192873709 p^{3} T^{15} + 2020488169 p^{4} T^{16} + 53384690 p^{5} T^{17} + 16564343 p^{6} T^{18} + 331941 p^{7} T^{19} + 102850 p^{8} T^{20} + 1279 p^{9} T^{21} + 440 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 29 T + 954 T^{2} - 19535 T^{3} + 391466 T^{4} - 6242427 T^{5} + 94941506 T^{6} - 1241853249 T^{7} + 15429769519 T^{8} - 170316190890 T^{9} + 1785874498212 T^{10} - 16877221454302 T^{11} + 151573995496588 T^{12} - 16877221454302 p T^{13} + 1785874498212 p^{2} T^{14} - 170316190890 p^{3} T^{15} + 15429769519 p^{4} T^{16} - 1241853249 p^{5} T^{17} + 94941506 p^{6} T^{18} - 6242427 p^{7} T^{19} + 391466 p^{8} T^{20} - 19535 p^{9} T^{21} + 954 p^{10} T^{22} - 29 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 2 T + 411 T^{2} - 2253 T^{3} + 89379 T^{4} - 639562 T^{5} + 14623097 T^{6} - 104340042 T^{7} + 1878227557 T^{8} - 12746093586 T^{9} + 193222566838 T^{10} - 1243919785871 T^{11} + 16586345035122 T^{12} - 1243919785871 p T^{13} + 193222566838 p^{2} T^{14} - 12746093586 p^{3} T^{15} + 1878227557 p^{4} T^{16} - 104340042 p^{5} T^{17} + 14623097 p^{6} T^{18} - 639562 p^{7} T^{19} + 89379 p^{8} T^{20} - 2253 p^{9} T^{21} + 411 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 26 T + 793 T^{2} - 14473 T^{3} + 269039 T^{4} - 3881982 T^{5} + 55392841 T^{6} - 676943158 T^{7} + 8110698807 T^{8} - 87408727066 T^{9} + 920478793140 T^{10} - 8929268159403 T^{11} + 84453485499158 T^{12} - 8929268159403 p T^{13} + 920478793140 p^{2} T^{14} - 87408727066 p^{3} T^{15} + 8110698807 p^{4} T^{16} - 676943158 p^{5} T^{17} + 55392841 p^{6} T^{18} - 3881982 p^{7} T^{19} + 269039 p^{8} T^{20} - 14473 p^{9} T^{21} + 793 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 41 T + 1575 T^{2} - 39952 T^{3} + 930020 T^{4} - 17532120 T^{5} + 305616583 T^{6} - 4610130597 T^{7} + 64782862031 T^{8} - 810703195874 T^{9} + 9496368505634 T^{10} - 100349127411920 T^{11} + 994856690864984 T^{12} - 100349127411920 p T^{13} + 9496368505634 p^{2} T^{14} - 810703195874 p^{3} T^{15} + 64782862031 p^{4} T^{16} - 4610130597 p^{5} T^{17} + 305616583 p^{6} T^{18} - 17532120 p^{7} T^{19} + 930020 p^{8} T^{20} - 39952 p^{9} T^{21} + 1575 p^{10} T^{22} - 41 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 7 T + 686 T^{2} + 3921 T^{3} + 226418 T^{4} + 1042781 T^{5} + 48755990 T^{6} + 178774699 T^{7} + 7795899247 T^{8} + 22845444702 T^{9} + 993525140156 T^{10} + 2436670240642 T^{11} + 105063768730428 T^{12} + 2436670240642 p T^{13} + 993525140156 p^{2} T^{14} + 22845444702 p^{3} T^{15} + 7795899247 p^{4} T^{16} + 178774699 p^{5} T^{17} + 48755990 p^{6} T^{18} + 1042781 p^{7} T^{19} + 226418 p^{8} T^{20} + 3921 p^{9} T^{21} + 686 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.61027285352085821487937943132, −2.48481097891978531670640045817, −2.37896428838326776495196291362, −2.31746650221425691113993224224, −2.26868303840845427713906837818, −2.14841326072457338430891470411, −2.12716079660360227964765324649, −2.11399341283601245350712208230, −2.05726273503079414713591679882, −2.00828835588433119446512108059, −1.90852729140628582202219519699, −1.76489806522869220371695722771, −1.59248687664043637863577571798, −1.47374920379272516696865367168, −1.40518332240939556255518970719, −1.36020333878090691801881071299, −1.27765851446567047280402261180, −1.23439166033214821112857177611, −1.05869620729122783900593829843, −1.02375406043033238903659347373, −0.947053740980286492538113080669, −0.846893167379349137781323212592, −0.73242954292581245761715919022, −0.66360615546239348215409642661, −0.58961682081734473335624635938, 0.58961682081734473335624635938, 0.66360615546239348215409642661, 0.73242954292581245761715919022, 0.846893167379349137781323212592, 0.947053740980286492538113080669, 1.02375406043033238903659347373, 1.05869620729122783900593829843, 1.23439166033214821112857177611, 1.27765851446567047280402261180, 1.36020333878090691801881071299, 1.40518332240939556255518970719, 1.47374920379272516696865367168, 1.59248687664043637863577571798, 1.76489806522869220371695722771, 1.90852729140628582202219519699, 2.00828835588433119446512108059, 2.05726273503079414713591679882, 2.11399341283601245350712208230, 2.12716079660360227964765324649, 2.14841326072457338430891470411, 2.26868303840845427713906837818, 2.31746650221425691113993224224, 2.37896428838326776495196291362, 2.48481097891978531670640045817, 2.61027285352085821487937943132

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

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