Properties

Label 24-4016e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.760\times 10^{43}$
Sign $1$
Analytic cond. $1.18263\times 10^{18}$
Root an. cond. $5.66285$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5·5-s − 5·7-s − 10·9-s − 10·11-s + 3·13-s − 15·15-s + 2·17-s − 15·19-s + 15·21-s − 20·23-s − 19·25-s + 37·27-s + 6·29-s − 14·31-s + 30·33-s − 25·35-s + 5·37-s − 9·39-s − 21·43-s − 50·45-s − 27·47-s − 36·49-s − 6·51-s + 22·53-s − 50·55-s + 45·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 2.23·5-s − 1.88·7-s − 3.33·9-s − 3.01·11-s + 0.832·13-s − 3.87·15-s + 0.485·17-s − 3.44·19-s + 3.27·21-s − 4.17·23-s − 3.79·25-s + 7.12·27-s + 1.11·29-s − 2.51·31-s + 5.22·33-s − 4.22·35-s + 0.821·37-s − 1.44·39-s − 3.20·43-s − 7.45·45-s − 3.93·47-s − 5.14·49-s − 0.840·51-s + 3.02·53-s − 6.74·55-s + 5.96·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 251^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 251^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 251^{12}\)
Sign: $1$
Analytic conductor: \(1.18263\times 10^{18}\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 2^{48} \cdot 251^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( ( 1 + T )^{12} \)
good3 \( 1 + p T + 19 T^{2} + 50 T^{3} + 190 T^{4} + 439 T^{5} + 1282 T^{6} + 2662 T^{7} + 2186 p T^{8} + 12352 T^{9} + 8915 p T^{10} + 45782 T^{11} + 88754 T^{12} + 45782 p T^{13} + 8915 p^{3} T^{14} + 12352 p^{3} T^{15} + 2186 p^{5} T^{16} + 2662 p^{5} T^{17} + 1282 p^{6} T^{18} + 439 p^{7} T^{19} + 190 p^{8} T^{20} + 50 p^{9} T^{21} + 19 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \)
5 \( 1 - p T + 44 T^{2} - 177 T^{3} + 928 T^{4} - 631 p T^{5} + 12519 T^{6} - 7373 p T^{7} + 120738 T^{8} - 312421 T^{9} + 879601 T^{10} - 2013517 T^{11} + 4972218 T^{12} - 2013517 p T^{13} + 879601 p^{2} T^{14} - 312421 p^{3} T^{15} + 120738 p^{4} T^{16} - 7373 p^{6} T^{17} + 12519 p^{6} T^{18} - 631 p^{8} T^{19} + 928 p^{8} T^{20} - 177 p^{9} T^{21} + 44 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \)
7 \( 1 + 5 T + 61 T^{2} + 234 T^{3} + 1656 T^{4} + 5163 T^{5} + 27663 T^{6} + 72782 T^{7} + 329230 T^{8} + 754003 T^{9} + 3052708 T^{10} + 6266814 T^{11} + 23313049 T^{12} + 6266814 p T^{13} + 3052708 p^{2} T^{14} + 754003 p^{3} T^{15} + 329230 p^{4} T^{16} + 72782 p^{5} T^{17} + 27663 p^{6} T^{18} + 5163 p^{7} T^{19} + 1656 p^{8} T^{20} + 234 p^{9} T^{21} + 61 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + 10 T + 140 T^{2} + 1003 T^{3} + 8199 T^{4} + 46419 T^{5} + 282004 T^{6} + 1327774 T^{7} + 6549093 T^{8} + 26384889 T^{9} + 10019560 p T^{10} + 385435349 T^{11} + 126546890 p T^{12} + 385435349 p T^{13} + 10019560 p^{3} T^{14} + 26384889 p^{3} T^{15} + 6549093 p^{4} T^{16} + 1327774 p^{5} T^{17} + 282004 p^{6} T^{18} + 46419 p^{7} T^{19} + 8199 p^{8} T^{20} + 1003 p^{9} T^{21} + 140 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 3 T + 99 T^{2} - 290 T^{3} + 4978 T^{4} - 13727 T^{5} + 165710 T^{6} - 422960 T^{7} + 310088 p T^{8} - 9416452 T^{9} + 75184897 T^{10} - 158659844 T^{11} + 1099925382 T^{12} - 158659844 p T^{13} + 75184897 p^{2} T^{14} - 9416452 p^{3} T^{15} + 310088 p^{5} T^{16} - 422960 p^{5} T^{17} + 165710 p^{6} T^{18} - 13727 p^{7} T^{19} + 4978 p^{8} T^{20} - 290 p^{9} T^{21} + 99 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 2 T + 141 T^{2} - 211 T^{3} + 9189 T^{4} - 8557 T^{5} + 369346 T^{6} - 123771 T^{7} + 10436566 T^{8} + 2186136 T^{9} + 228044014 T^{10} + 123426362 T^{11} + 245148867 p T^{12} + 123426362 p T^{13} + 228044014 p^{2} T^{14} + 2186136 p^{3} T^{15} + 10436566 p^{4} T^{16} - 123771 p^{5} T^{17} + 369346 p^{6} T^{18} - 8557 p^{7} T^{19} + 9189 p^{8} T^{20} - 211 p^{9} T^{21} + 141 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 15 T + 275 T^{2} + 2854 T^{3} + 31018 T^{4} + 249838 T^{5} + 2022975 T^{6} + 13349013 T^{7} + 87267683 T^{8} + 485194285 T^{9} + 2656933750 T^{10} + 12617725635 T^{11} + 58890558116 T^{12} + 12617725635 p T^{13} + 2656933750 p^{2} T^{14} + 485194285 p^{3} T^{15} + 87267683 p^{4} T^{16} + 13349013 p^{5} T^{17} + 2022975 p^{6} T^{18} + 249838 p^{7} T^{19} + 31018 p^{8} T^{20} + 2854 p^{9} T^{21} + 275 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 20 T + 365 T^{2} + 192 p T^{3} + 48516 T^{4} + 432668 T^{5} + 3561459 T^{6} + 25491272 T^{7} + 170978142 T^{8} + 1030965126 T^{9} + 5896418020 T^{10} + 30856094338 T^{11} + 154253916053 T^{12} + 30856094338 p T^{13} + 5896418020 p^{2} T^{14} + 1030965126 p^{3} T^{15} + 170978142 p^{4} T^{16} + 25491272 p^{5} T^{17} + 3561459 p^{6} T^{18} + 432668 p^{7} T^{19} + 48516 p^{8} T^{20} + 192 p^{10} T^{21} + 365 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 6 T + 263 T^{2} - 1464 T^{3} + 32996 T^{4} - 170821 T^{5} + 2623910 T^{6} - 12596328 T^{7} + 148094014 T^{8} - 653381511 T^{9} + 6282380187 T^{10} - 25073897238 T^{11} + 206197153794 T^{12} - 25073897238 p T^{13} + 6282380187 p^{2} T^{14} - 653381511 p^{3} T^{15} + 148094014 p^{4} T^{16} - 12596328 p^{5} T^{17} + 2623910 p^{6} T^{18} - 170821 p^{7} T^{19} + 32996 p^{8} T^{20} - 1464 p^{9} T^{21} + 263 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 14 T + 307 T^{2} + 3043 T^{3} + 39363 T^{4} + 309078 T^{5} + 3054515 T^{6} + 20375941 T^{7} + 170212169 T^{8} + 1004678182 T^{9} + 7390624625 T^{10} + 39099308202 T^{11} + 256163422028 T^{12} + 39099308202 p T^{13} + 7390624625 p^{2} T^{14} + 1004678182 p^{3} T^{15} + 170212169 p^{4} T^{16} + 20375941 p^{5} T^{17} + 3054515 p^{6} T^{18} + 309078 p^{7} T^{19} + 39363 p^{8} T^{20} + 3043 p^{9} T^{21} + 307 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 5 T + 312 T^{2} - 1415 T^{3} + 48022 T^{4} - 198367 T^{5} + 4789573 T^{6} - 17999713 T^{7} + 343463566 T^{8} - 1167787725 T^{9} + 18608835879 T^{10} - 56684284147 T^{11} + 780021299974 T^{12} - 56684284147 p T^{13} + 18608835879 p^{2} T^{14} - 1167787725 p^{3} T^{15} + 343463566 p^{4} T^{16} - 17999713 p^{5} T^{17} + 4789573 p^{6} T^{18} - 198367 p^{7} T^{19} + 48022 p^{8} T^{20} - 1415 p^{9} T^{21} + 312 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 258 T^{2} + 473 T^{3} + 33090 T^{4} + 98836 T^{5} + 2993547 T^{6} + 9926755 T^{7} + 211013443 T^{8} + 685203796 T^{9} + 11761926780 T^{10} + 36494399132 T^{11} + 530959735377 T^{12} + 36494399132 p T^{13} + 11761926780 p^{2} T^{14} + 685203796 p^{3} T^{15} + 211013443 p^{4} T^{16} + 9926755 p^{5} T^{17} + 2993547 p^{6} T^{18} + 98836 p^{7} T^{19} + 33090 p^{8} T^{20} + 473 p^{9} T^{21} + 258 p^{10} T^{22} + p^{12} T^{24} \)
43 \( 1 + 21 T + 566 T^{2} + 8605 T^{3} + 137410 T^{4} + 1651519 T^{5} + 19709307 T^{6} + 195788145 T^{7} + 1892909276 T^{8} + 15920662733 T^{9} + 129322691089 T^{10} + 932042200511 T^{11} + 6470033718086 T^{12} + 932042200511 p T^{13} + 129322691089 p^{2} T^{14} + 15920662733 p^{3} T^{15} + 1892909276 p^{4} T^{16} + 195788145 p^{5} T^{17} + 19709307 p^{6} T^{18} + 1651519 p^{7} T^{19} + 137410 p^{8} T^{20} + 8605 p^{9} T^{21} + 566 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 27 T + 721 T^{2} + 12646 T^{3} + 205206 T^{4} + 2719622 T^{5} + 33303256 T^{6} + 356780847 T^{7} + 3554349185 T^{8} + 31875619896 T^{9} + 267301104519 T^{10} + 2043077919444 T^{11} + 14644952368176 T^{12} + 2043077919444 p T^{13} + 267301104519 p^{2} T^{14} + 31875619896 p^{3} T^{15} + 3554349185 p^{4} T^{16} + 356780847 p^{5} T^{17} + 33303256 p^{6} T^{18} + 2719622 p^{7} T^{19} + 205206 p^{8} T^{20} + 12646 p^{9} T^{21} + 721 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 22 T + 457 T^{2} - 6131 T^{3} + 81712 T^{4} - 864884 T^{5} + 9238431 T^{6} - 83791862 T^{7} + 777068847 T^{8} - 6324907486 T^{9} + 53066638192 T^{10} - 395846312399 T^{11} + 3050517811680 T^{12} - 395846312399 p T^{13} + 53066638192 p^{2} T^{14} - 6324907486 p^{3} T^{15} + 777068847 p^{4} T^{16} - 83791862 p^{5} T^{17} + 9238431 p^{6} T^{18} - 864884 p^{7} T^{19} + 81712 p^{8} T^{20} - 6131 p^{9} T^{21} + 457 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 23 T + 463 T^{2} + 5991 T^{3} + 76539 T^{4} + 797165 T^{5} + 8616593 T^{6} + 80045466 T^{7} + 761614967 T^{8} + 6355022897 T^{9} + 55110088864 T^{10} + 428918254928 T^{11} + 3481229277834 T^{12} + 428918254928 p T^{13} + 55110088864 p^{2} T^{14} + 6355022897 p^{3} T^{15} + 761614967 p^{4} T^{16} + 80045466 p^{5} T^{17} + 8616593 p^{6} T^{18} + 797165 p^{7} T^{19} + 76539 p^{8} T^{20} + 5991 p^{9} T^{21} + 463 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 4 T + 515 T^{2} - 2183 T^{3} + 126825 T^{4} - 553350 T^{5} + 20068525 T^{6} - 86790548 T^{7} + 2298315906 T^{8} - 9463792069 T^{9} + 201443760970 T^{10} - 761468440966 T^{11} + 13837315630820 T^{12} - 761468440966 p T^{13} + 201443760970 p^{2} T^{14} - 9463792069 p^{3} T^{15} + 2298315906 p^{4} T^{16} - 86790548 p^{5} T^{17} + 20068525 p^{6} T^{18} - 553350 p^{7} T^{19} + 126825 p^{8} T^{20} - 2183 p^{9} T^{21} + 515 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 26 T + 734 T^{2} + 12241 T^{3} + 194371 T^{4} + 2301401 T^{5} + 24936552 T^{6} + 208315088 T^{7} + 1508590671 T^{8} + 6757901265 T^{9} + 10934295138 T^{10} - 306038227373 T^{11} - 3111488124494 T^{12} - 306038227373 p T^{13} + 10934295138 p^{2} T^{14} + 6757901265 p^{3} T^{15} + 1508590671 p^{4} T^{16} + 208315088 p^{5} T^{17} + 24936552 p^{6} T^{18} + 2301401 p^{7} T^{19} + 194371 p^{8} T^{20} + 12241 p^{9} T^{21} + 734 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 23 T + 815 T^{2} + 13944 T^{3} + 290174 T^{4} + 4016108 T^{5} + 62482250 T^{6} + 729493421 T^{7} + 9247078025 T^{8} + 93075600870 T^{9} + 1000403940743 T^{10} + 8767603232972 T^{11} + 81491494383600 T^{12} + 8767603232972 p T^{13} + 1000403940743 p^{2} T^{14} + 93075600870 p^{3} T^{15} + 9247078025 p^{4} T^{16} + 729493421 p^{5} T^{17} + 62482250 p^{6} T^{18} + 4016108 p^{7} T^{19} + 290174 p^{8} T^{20} + 13944 p^{9} T^{21} + 815 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 + 8 T + 635 T^{2} + 4195 T^{3} + 189057 T^{4} + 1034616 T^{5} + 35458617 T^{6} + 161383993 T^{7} + 4745268235 T^{8} + 18194832974 T^{9} + 485587865335 T^{10} + 1612273953570 T^{11} + 39501142360716 T^{12} + 1612273953570 p T^{13} + 485587865335 p^{2} T^{14} + 18194832974 p^{3} T^{15} + 4745268235 p^{4} T^{16} + 161383993 p^{5} T^{17} + 35458617 p^{6} T^{18} + 1034616 p^{7} T^{19} + 189057 p^{8} T^{20} + 4195 p^{9} T^{21} + 635 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 37 T + 1170 T^{2} + 26323 T^{3} + 524294 T^{4} + 8813126 T^{5} + 134846261 T^{6} + 1837624125 T^{7} + 23158328723 T^{8} + 266053942831 T^{9} + 2851382404096 T^{10} + 28152678898468 T^{11} + 260296843460799 T^{12} + 28152678898468 p T^{13} + 2851382404096 p^{2} T^{14} + 266053942831 p^{3} T^{15} + 23158328723 p^{4} T^{16} + 1837624125 p^{5} T^{17} + 134846261 p^{6} T^{18} + 8813126 p^{7} T^{19} + 524294 p^{8} T^{20} + 26323 p^{9} T^{21} + 1170 p^{10} T^{22} + 37 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 30 T + 998 T^{2} + 19521 T^{3} + 386595 T^{4} + 5774396 T^{5} + 86624835 T^{6} + 1074210737 T^{7} + 13455746937 T^{8} + 145378443767 T^{9} + 1590934546399 T^{10} + 15272101715923 T^{11} + 148277925367526 T^{12} + 15272101715923 p T^{13} + 1590934546399 p^{2} T^{14} + 145378443767 p^{3} T^{15} + 13455746937 p^{4} T^{16} + 1074210737 p^{5} T^{17} + 86624835 p^{6} T^{18} + 5774396 p^{7} T^{19} + 386595 p^{8} T^{20} + 19521 p^{9} T^{21} + 998 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 3 T + 695 T^{2} - 656 T^{3} + 229968 T^{4} + 161447 T^{5} + 49555045 T^{6} + 88908024 T^{7} + 7925590920 T^{8} + 18070410375 T^{9} + 993701773490 T^{10} + 2286483477062 T^{11} + 99101678422277 T^{12} + 2286483477062 p T^{13} + 993701773490 p^{2} T^{14} + 18070410375 p^{3} T^{15} + 7925590920 p^{4} T^{16} + 88908024 p^{5} T^{17} + 49555045 p^{6} T^{18} + 161447 p^{7} T^{19} + 229968 p^{8} T^{20} - 656 p^{9} T^{21} + 695 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 4 T + 705 T^{2} + 2332 T^{3} + 244683 T^{4} + 665112 T^{5} + 55974764 T^{6} + 126126264 T^{7} + 9477471873 T^{8} + 18038171780 T^{9} + 1258847536915 T^{10} + 2084337404684 T^{11} + 135161661133350 T^{12} + 2084337404684 p T^{13} + 1258847536915 p^{2} T^{14} + 18038171780 p^{3} T^{15} + 9477471873 p^{4} T^{16} + 126126264 p^{5} T^{17} + 55974764 p^{6} T^{18} + 665112 p^{7} T^{19} + 244683 p^{8} T^{20} + 2332 p^{9} T^{21} + 705 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
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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

−3.06928644590553552026111129050, −2.81274810302788273356405306013, −2.79024197409501247673424278457, −2.65595391378646636224193591514, −2.56410964652605455620336454075, −2.52833493871770255678974977118, −2.48259673865684132913672168348, −2.44900483896302482728503846061, −2.42749690499505018682263033789, −2.31635925449867468372808163276, −2.28198109815270128013284526751, −2.13715427569300537192239170735, −2.11970510348236414701939609538, −2.04544835812598832916041014582, −1.80045252670244753853141659915, −1.68219006718632468243267875047, −1.66944269580458829375234502453, −1.55952005505326650046201567769, −1.49780465102691307151329616483, −1.46943544490769101040674591048, −1.36198024176726468282318917597, −1.29975186323041556279854385929, −1.18676793956537070322908462596, −1.10378494981113955925549923324, −0.928250277803438619434796091969, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.928250277803438619434796091969, 1.10378494981113955925549923324, 1.18676793956537070322908462596, 1.29975186323041556279854385929, 1.36198024176726468282318917597, 1.46943544490769101040674591048, 1.49780465102691307151329616483, 1.55952005505326650046201567769, 1.66944269580458829375234502453, 1.68219006718632468243267875047, 1.80045252670244753853141659915, 2.04544835812598832916041014582, 2.11970510348236414701939609538, 2.13715427569300537192239170735, 2.28198109815270128013284526751, 2.31635925449867468372808163276, 2.42749690499505018682263033789, 2.44900483896302482728503846061, 2.48259673865684132913672168348, 2.52833493871770255678974977118, 2.56410964652605455620336454075, 2.65595391378646636224193591514, 2.79024197409501247673424278457, 2.81274810302788273356405306013, 3.06928644590553552026111129050

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

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