Properties

Degree 64
Conductor $ 3^{64} \cdot 7^{32} \cdot 127^{32} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 32

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 17·4-s − 32·7-s − 14·13-s + 134·16-s − 30·19-s − 62·25-s + 544·28-s − 58·31-s + 8·37-s + 6·43-s + 528·49-s + 238·52-s − 46·61-s − 643·64-s − 8·67-s − 60·73-s + 510·76-s − 74·79-s + 448·91-s − 44·97-s + 1.05e3·100-s − 60·103-s − 52·109-s − 4.28e3·112-s − 167·121-s + 986·124-s + 127-s + ⋯
L(s)  = 1  − 8.5·4-s − 12.0·7-s − 3.88·13-s + 67/2·16-s − 6.88·19-s − 12.3·25-s + 102.·28-s − 10.4·31-s + 1.31·37-s + 0.914·43-s + 75.4·49-s + 33.0·52-s − 5.88·61-s − 80.3·64-s − 0.977·67-s − 7.02·73-s + 58.5·76-s − 8.32·79-s + 46.9·91-s − 4.46·97-s + 105.·100-s − 5.91·103-s − 4.98·109-s − 405.·112-s − 15.1·121-s + 88.5·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{64} \cdot 7^{32} \cdot 127^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{64} \cdot 7^{32} \cdot 127^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{32} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(3^{64} \cdot 7^{32} \cdot 127^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  32
Selberg data  =  $(64,\ 3^{64} \cdot 7^{32} \cdot 127^{32} ,\ ( \ : [1/2]^{32} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \(F_p\) is a polynomial of degree 64. If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 63.
$p$$F_p$
bad3 \( 1 \)
7 \( ( 1 + T )^{32} \)
127 \( ( 1 + T )^{32} \)
good2 \( 1 + 17 T^{2} + 155 T^{4} + 125 p^{3} T^{6} + 5123 T^{8} + 11097 p T^{10} + 84503 T^{12} + 144939 p T^{14} + 113869 p^{3} T^{16} + 2653847 T^{18} + 7232849 T^{20} + 9288351 p T^{22} + 45229899 T^{24} + 13111457 p^{3} T^{26} + 14534373 p^{4} T^{28} + 123553503 p^{2} T^{30} + 63028041 p^{4} T^{32} + 123553503 p^{4} T^{34} + 14534373 p^{8} T^{36} + 13111457 p^{9} T^{38} + 45229899 p^{8} T^{40} + 9288351 p^{11} T^{42} + 7232849 p^{12} T^{44} + 2653847 p^{14} T^{46} + 113869 p^{19} T^{48} + 144939 p^{19} T^{50} + 84503 p^{20} T^{52} + 11097 p^{23} T^{54} + 5123 p^{24} T^{56} + 125 p^{29} T^{58} + 155 p^{28} T^{60} + 17 p^{30} T^{62} + p^{32} T^{64} \)
5 \( 1 + 62 T^{2} + 1977 T^{4} + 42846 T^{6} + 704707 T^{8} + 9319788 T^{10} + 102553319 T^{12} + 958881781 T^{14} + 7711275388 T^{16} + 53588419309 T^{18} + 320648455553 T^{20} + 1625118169584 T^{22} + 6677393171933 T^{24} + 19458462902206 T^{26} + 2964028145267 p T^{28} - 258220040883296 T^{30} - 1927224259903354 T^{32} - 258220040883296 p^{2} T^{34} + 2964028145267 p^{5} T^{36} + 19458462902206 p^{6} T^{38} + 6677393171933 p^{8} T^{40} + 1625118169584 p^{10} T^{42} + 320648455553 p^{12} T^{44} + 53588419309 p^{14} T^{46} + 7711275388 p^{16} T^{48} + 958881781 p^{18} T^{50} + 102553319 p^{20} T^{52} + 9319788 p^{22} T^{54} + 704707 p^{24} T^{56} + 42846 p^{26} T^{58} + 1977 p^{28} T^{60} + 62 p^{30} T^{62} + p^{32} T^{64} \)
11 \( 1 + 167 T^{2} + 13927 T^{4} + 772703 T^{6} + 2914482 p T^{8} + 96361893 p T^{10} + 29062714667 T^{12} + 679001059672 T^{14} + 13787640798818 T^{16} + 247086891554488 T^{18} + 3958523329848232 T^{20} + 57380415900086043 T^{22} + 762225633953779080 T^{24} + 9418378494637864487 T^{26} + \)\(11\!\cdots\!50\)\( T^{28} + \)\(12\!\cdots\!85\)\( T^{30} + \)\(13\!\cdots\!46\)\( T^{32} + \)\(12\!\cdots\!85\)\( p^{2} T^{34} + \)\(11\!\cdots\!50\)\( p^{4} T^{36} + 9418378494637864487 p^{6} T^{38} + 762225633953779080 p^{8} T^{40} + 57380415900086043 p^{10} T^{42} + 3958523329848232 p^{12} T^{44} + 247086891554488 p^{14} T^{46} + 13787640798818 p^{16} T^{48} + 679001059672 p^{18} T^{50} + 29062714667 p^{20} T^{52} + 96361893 p^{23} T^{54} + 2914482 p^{25} T^{56} + 772703 p^{26} T^{58} + 13927 p^{28} T^{60} + 167 p^{30} T^{62} + p^{32} T^{64} \)
13 \( ( 1 + 7 T + 125 T^{2} + 638 T^{3} + 6806 T^{4} + 26922 T^{5} + 225887 T^{6} + 698283 T^{7} + 5261978 T^{8} + 12335079 T^{9} + 94086260 T^{10} + 155962343 T^{11} + 1398436464 T^{12} + 113840167 p T^{13} + 18751899800 T^{14} + 12720709761 T^{15} + 243845718462 T^{16} + 12720709761 p T^{17} + 18751899800 p^{2} T^{18} + 113840167 p^{4} T^{19} + 1398436464 p^{4} T^{20} + 155962343 p^{5} T^{21} + 94086260 p^{6} T^{22} + 12335079 p^{7} T^{23} + 5261978 p^{8} T^{24} + 698283 p^{9} T^{25} + 225887 p^{10} T^{26} + 26922 p^{11} T^{27} + 6806 p^{12} T^{28} + 638 p^{13} T^{29} + 125 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
17 \( 1 + 246 T^{2} + 1724 p T^{4} + 133319 p T^{6} + 128983355 T^{8} + 5821033050 T^{10} + 219482813950 T^{12} + 7187431001960 T^{14} + 210334257203010 T^{16} + 5612442266673033 T^{18} + 138447284643409455 T^{20} + 3187206419435650282 T^{22} + 68924736567233049258 T^{24} + \)\(14\!\cdots\!74\)\( T^{26} + \)\(27\!\cdots\!13\)\( T^{28} + \)\(49\!\cdots\!20\)\( T^{30} + \)\(86\!\cdots\!68\)\( T^{32} + \)\(49\!\cdots\!20\)\( p^{2} T^{34} + \)\(27\!\cdots\!13\)\( p^{4} T^{36} + \)\(14\!\cdots\!74\)\( p^{6} T^{38} + 68924736567233049258 p^{8} T^{40} + 3187206419435650282 p^{10} T^{42} + 138447284643409455 p^{12} T^{44} + 5612442266673033 p^{14} T^{46} + 210334257203010 p^{16} T^{48} + 7187431001960 p^{18} T^{50} + 219482813950 p^{20} T^{52} + 5821033050 p^{22} T^{54} + 128983355 p^{24} T^{56} + 133319 p^{27} T^{58} + 1724 p^{29} T^{60} + 246 p^{30} T^{62} + p^{32} T^{64} \)
19 \( ( 1 + 15 T + 263 T^{2} + 2710 T^{3} + 28653 T^{4} + 229391 T^{5} + 1836708 T^{6} + 12155774 T^{7} + 80311567 T^{8} + 24053526 p T^{9} + 137706049 p T^{10} + 13209134013 T^{11} + 68030532395 T^{12} + 314053096114 T^{13} + 1504326969994 T^{14} + 6535557195209 T^{15} + 1570617460736 p T^{16} + 6535557195209 p T^{17} + 1504326969994 p^{2} T^{18} + 314053096114 p^{3} T^{19} + 68030532395 p^{4} T^{20} + 13209134013 p^{5} T^{21} + 137706049 p^{7} T^{22} + 24053526 p^{8} T^{23} + 80311567 p^{8} T^{24} + 12155774 p^{9} T^{25} + 1836708 p^{10} T^{26} + 229391 p^{11} T^{27} + 28653 p^{12} T^{28} + 2710 p^{13} T^{29} + 263 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
23 \( 1 + 254 T^{2} + 34943 T^{4} + 3393069 T^{6} + 258597088 T^{8} + 711734694 p T^{10} + 891587440800 T^{12} + 42792734910028 T^{14} + 1841442418669100 T^{16} + 71966350132913380 T^{18} + 2579648360979918710 T^{20} + 85454309425736767400 T^{22} + \)\(26\!\cdots\!28\)\( T^{24} + \)\(75\!\cdots\!29\)\( T^{26} + \)\(20\!\cdots\!19\)\( T^{28} + \)\(51\!\cdots\!22\)\( T^{30} + \)\(12\!\cdots\!10\)\( T^{32} + \)\(51\!\cdots\!22\)\( p^{2} T^{34} + \)\(20\!\cdots\!19\)\( p^{4} T^{36} + \)\(75\!\cdots\!29\)\( p^{6} T^{38} + \)\(26\!\cdots\!28\)\( p^{8} T^{40} + 85454309425736767400 p^{10} T^{42} + 2579648360979918710 p^{12} T^{44} + 71966350132913380 p^{14} T^{46} + 1841442418669100 p^{16} T^{48} + 42792734910028 p^{18} T^{50} + 891587440800 p^{20} T^{52} + 711734694 p^{23} T^{54} + 258597088 p^{24} T^{56} + 3393069 p^{26} T^{58} + 34943 p^{28} T^{60} + 254 p^{30} T^{62} + p^{32} T^{64} \)
29 \( 1 + 466 T^{2} + 107016 T^{4} + 16128520 T^{6} + 1793116898 T^{8} + 156813160784 T^{10} + 11241735817303 T^{12} + 680510125734998 T^{14} + 35610164077596639 T^{16} + 1643794623308716326 T^{18} + 68199327965360587652 T^{20} + \)\(25\!\cdots\!52\)\( T^{22} + \)\(91\!\cdots\!24\)\( T^{24} + \)\(30\!\cdots\!12\)\( T^{26} + \)\(96\!\cdots\!21\)\( T^{28} + \)\(29\!\cdots\!02\)\( T^{30} + \)\(87\!\cdots\!48\)\( T^{32} + \)\(29\!\cdots\!02\)\( p^{2} T^{34} + \)\(96\!\cdots\!21\)\( p^{4} T^{36} + \)\(30\!\cdots\!12\)\( p^{6} T^{38} + \)\(91\!\cdots\!24\)\( p^{8} T^{40} + \)\(25\!\cdots\!52\)\( p^{10} T^{42} + 68199327965360587652 p^{12} T^{44} + 1643794623308716326 p^{14} T^{46} + 35610164077596639 p^{16} T^{48} + 680510125734998 p^{18} T^{50} + 11241735817303 p^{20} T^{52} + 156813160784 p^{22} T^{54} + 1793116898 p^{24} T^{56} + 16128520 p^{26} T^{58} + 107016 p^{28} T^{60} + 466 p^{30} T^{62} + p^{32} T^{64} \)
31 \( ( 1 + 29 T + 663 T^{2} + 10631 T^{3} + 149270 T^{4} + 1758263 T^{5} + 18941695 T^{6} + 182082745 T^{7} + 1637775352 T^{8} + 13535765660 T^{9} + 3420370090 p T^{10} + 775193410037 T^{11} + 5411216800890 T^{12} + 35557797916839 T^{13} + 224009022151708 T^{14} + 1334389874382274 T^{15} + 7635131509861374 T^{16} + 1334389874382274 p T^{17} + 224009022151708 p^{2} T^{18} + 35557797916839 p^{3} T^{19} + 5411216800890 p^{4} T^{20} + 775193410037 p^{5} T^{21} + 3420370090 p^{7} T^{22} + 13535765660 p^{7} T^{23} + 1637775352 p^{8} T^{24} + 182082745 p^{9} T^{25} + 18941695 p^{10} T^{26} + 1758263 p^{11} T^{27} + 149270 p^{12} T^{28} + 10631 p^{13} T^{29} + 663 p^{14} T^{30} + 29 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
37 \( ( 1 - 4 T + 255 T^{2} - 455 T^{3} + 31552 T^{4} + 6504 T^{5} + 2683663 T^{6} + 5121125 T^{7} + 183948154 T^{8} + 573570658 T^{9} + 10874974419 T^{10} + 40794180936 T^{11} + 564307721271 T^{12} + 59780099455 p T^{13} + 25676142242884 T^{14} + 97713139793825 T^{15} + 1018830346934130 T^{16} + 97713139793825 p T^{17} + 25676142242884 p^{2} T^{18} + 59780099455 p^{4} T^{19} + 564307721271 p^{4} T^{20} + 40794180936 p^{5} T^{21} + 10874974419 p^{6} T^{22} + 573570658 p^{7} T^{23} + 183948154 p^{8} T^{24} + 5121125 p^{9} T^{25} + 2683663 p^{10} T^{26} + 6504 p^{11} T^{27} + 31552 p^{12} T^{28} - 455 p^{13} T^{29} + 255 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
41 \( 1 + 710 T^{2} + 247844 T^{4} + 56758103 T^{6} + 9604569470 T^{8} + 1283125187044 T^{10} + 141258701651661 T^{12} + 13211970956545464 T^{14} + 1074422854313327366 T^{16} + 77374962692186717263 T^{18} + \)\(50\!\cdots\!21\)\( T^{20} + \)\(29\!\cdots\!72\)\( T^{22} + \)\(16\!\cdots\!80\)\( T^{24} + \)\(80\!\cdots\!58\)\( T^{26} + \)\(38\!\cdots\!68\)\( T^{28} + \)\(16\!\cdots\!18\)\( T^{30} + \)\(71\!\cdots\!18\)\( T^{32} + \)\(16\!\cdots\!18\)\( p^{2} T^{34} + \)\(38\!\cdots\!68\)\( p^{4} T^{36} + \)\(80\!\cdots\!58\)\( p^{6} T^{38} + \)\(16\!\cdots\!80\)\( p^{8} T^{40} + \)\(29\!\cdots\!72\)\( p^{10} T^{42} + \)\(50\!\cdots\!21\)\( p^{12} T^{44} + 77374962692186717263 p^{14} T^{46} + 1074422854313327366 p^{16} T^{48} + 13211970956545464 p^{18} T^{50} + 141258701651661 p^{20} T^{52} + 1283125187044 p^{22} T^{54} + 9604569470 p^{24} T^{56} + 56758103 p^{26} T^{58} + 247844 p^{28} T^{60} + 710 p^{30} T^{62} + p^{32} T^{64} \)
43 \( ( 1 - 3 T + 280 T^{2} - 900 T^{3} + 43371 T^{4} - 136136 T^{5} + 4785595 T^{6} - 14225659 T^{7} + 413957508 T^{8} - 1155914307 T^{9} + 29568858839 T^{10} - 77395574352 T^{11} + 1797290381613 T^{12} - 4409573562716 T^{13} + 94565381045758 T^{14} - 217524522906259 T^{15} + 4345162616769766 T^{16} - 217524522906259 p T^{17} + 94565381045758 p^{2} T^{18} - 4409573562716 p^{3} T^{19} + 1797290381613 p^{4} T^{20} - 77395574352 p^{5} T^{21} + 29568858839 p^{6} T^{22} - 1155914307 p^{7} T^{23} + 413957508 p^{8} T^{24} - 14225659 p^{9} T^{25} + 4785595 p^{10} T^{26} - 136136 p^{11} T^{27} + 43371 p^{12} T^{28} - 900 p^{13} T^{29} + 280 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
47 \( 1 + 679 T^{2} + 232059 T^{4} + 53150823 T^{6} + 9173783831 T^{8} + 1272882648402 T^{10} + 147968100356606 T^{12} + 14833212549872526 T^{14} + 1310149625895131335 T^{16} + \)\(10\!\cdots\!75\)\( T^{18} + \)\(74\!\cdots\!53\)\( T^{20} + \)\(49\!\cdots\!09\)\( T^{22} + \)\(30\!\cdots\!91\)\( T^{24} + \)\(17\!\cdots\!30\)\( T^{26} + \)\(92\!\cdots\!62\)\( T^{28} + \)\(46\!\cdots\!80\)\( T^{30} + \)\(22\!\cdots\!48\)\( T^{32} + \)\(46\!\cdots\!80\)\( p^{2} T^{34} + \)\(92\!\cdots\!62\)\( p^{4} T^{36} + \)\(17\!\cdots\!30\)\( p^{6} T^{38} + \)\(30\!\cdots\!91\)\( p^{8} T^{40} + \)\(49\!\cdots\!09\)\( p^{10} T^{42} + \)\(74\!\cdots\!53\)\( p^{12} T^{44} + \)\(10\!\cdots\!75\)\( p^{14} T^{46} + 1310149625895131335 p^{16} T^{48} + 14833212549872526 p^{18} T^{50} + 147968100356606 p^{20} T^{52} + 1272882648402 p^{22} T^{54} + 9173783831 p^{24} T^{56} + 53150823 p^{26} T^{58} + 232059 p^{28} T^{60} + 679 p^{30} T^{62} + p^{32} T^{64} \)
53 \( 1 + 802 T^{2} + 318665 T^{4} + 83582816 T^{6} + 16278529752 T^{8} + 2512622445575 T^{10} + 320678415721989 T^{12} + 34903621541205957 T^{14} + 3320832006010143663 T^{16} + \)\(28\!\cdots\!48\)\( T^{18} + \)\(21\!\cdots\!70\)\( T^{20} + \)\(15\!\cdots\!60\)\( T^{22} + \)\(10\!\cdots\!90\)\( T^{24} + \)\(65\!\cdots\!57\)\( T^{26} + \)\(39\!\cdots\!72\)\( T^{28} + \)\(22\!\cdots\!85\)\( T^{30} + \)\(12\!\cdots\!52\)\( T^{32} + \)\(22\!\cdots\!85\)\( p^{2} T^{34} + \)\(39\!\cdots\!72\)\( p^{4} T^{36} + \)\(65\!\cdots\!57\)\( p^{6} T^{38} + \)\(10\!\cdots\!90\)\( p^{8} T^{40} + \)\(15\!\cdots\!60\)\( p^{10} T^{42} + \)\(21\!\cdots\!70\)\( p^{12} T^{44} + \)\(28\!\cdots\!48\)\( p^{14} T^{46} + 3320832006010143663 p^{16} T^{48} + 34903621541205957 p^{18} T^{50} + 320678415721989 p^{20} T^{52} + 2512622445575 p^{22} T^{54} + 16278529752 p^{24} T^{56} + 83582816 p^{26} T^{58} + 318665 p^{28} T^{60} + 802 p^{30} T^{62} + p^{32} T^{64} \)
59 \( 1 + 1120 T^{2} + 622335 T^{4} + 228455737 T^{6} + 62274427788 T^{8} + 13438035936238 T^{10} + 2390684029020848 T^{12} + 360717983198457514 T^{14} + 47144666981047435432 T^{16} + \)\(54\!\cdots\!54\)\( T^{18} + \)\(55\!\cdots\!50\)\( T^{20} + \)\(51\!\cdots\!52\)\( T^{22} + \)\(43\!\cdots\!20\)\( T^{24} + \)\(33\!\cdots\!65\)\( T^{26} + \)\(24\!\cdots\!55\)\( T^{28} + \)\(15\!\cdots\!32\)\( T^{30} + \)\(97\!\cdots\!62\)\( T^{32} + \)\(15\!\cdots\!32\)\( p^{2} T^{34} + \)\(24\!\cdots\!55\)\( p^{4} T^{36} + \)\(33\!\cdots\!65\)\( p^{6} T^{38} + \)\(43\!\cdots\!20\)\( p^{8} T^{40} + \)\(51\!\cdots\!52\)\( p^{10} T^{42} + \)\(55\!\cdots\!50\)\( p^{12} T^{44} + \)\(54\!\cdots\!54\)\( p^{14} T^{46} + 47144666981047435432 p^{16} T^{48} + 360717983198457514 p^{18} T^{50} + 2390684029020848 p^{20} T^{52} + 13438035936238 p^{22} T^{54} + 62274427788 p^{24} T^{56} + 228455737 p^{26} T^{58} + 622335 p^{28} T^{60} + 1120 p^{30} T^{62} + p^{32} T^{64} \)
61 \( ( 1 + 23 T + 539 T^{2} + 7687 T^{3} + 112458 T^{4} + 1222285 T^{5} + 13884757 T^{6} + 122270937 T^{7} + 1143433038 T^{8} + 7863900970 T^{9} + 58421778250 T^{10} + 228049117003 T^{11} + 785585832768 T^{12} - 13004212152489 T^{13} - 152684740454644 T^{14} - 2078435290981404 T^{15} - 14744717849330094 T^{16} - 2078435290981404 p T^{17} - 152684740454644 p^{2} T^{18} - 13004212152489 p^{3} T^{19} + 785585832768 p^{4} T^{20} + 228049117003 p^{5} T^{21} + 58421778250 p^{6} T^{22} + 7863900970 p^{7} T^{23} + 1143433038 p^{8} T^{24} + 122270937 p^{9} T^{25} + 13884757 p^{10} T^{26} + 1222285 p^{11} T^{27} + 112458 p^{12} T^{28} + 7687 p^{13} T^{29} + 539 p^{14} T^{30} + 23 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
67 \( ( 1 + 4 T + 522 T^{2} + 2434 T^{3} + 145264 T^{4} + 732647 T^{5} + 28128633 T^{6} + 146870117 T^{7} + 4190928052 T^{8} + 21980039885 T^{9} + 505176581085 T^{10} + 2600865812823 T^{11} + 50679968060488 T^{12} + 251061871173894 T^{13} + 4301568819917752 T^{14} + 20112989425451304 T^{15} + 311565455607704678 T^{16} + 20112989425451304 p T^{17} + 4301568819917752 p^{2} T^{18} + 251061871173894 p^{3} T^{19} + 50679968060488 p^{4} T^{20} + 2600865812823 p^{5} T^{21} + 505176581085 p^{6} T^{22} + 21980039885 p^{7} T^{23} + 4190928052 p^{8} T^{24} + 146870117 p^{9} T^{25} + 28128633 p^{10} T^{26} + 732647 p^{11} T^{27} + 145264 p^{12} T^{28} + 2434 p^{13} T^{29} + 522 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
71 \( 1 + 918 T^{2} + 434174 T^{4} + 140521605 T^{6} + 34928550698 T^{8} + 7100784349181 T^{10} + 1228366591907076 T^{12} + 185788521405772448 T^{14} + 25052489398240412203 T^{16} + \)\(30\!\cdots\!63\)\( T^{18} + \)\(34\!\cdots\!53\)\( T^{20} + \)\(35\!\cdots\!00\)\( T^{22} + \)\(33\!\cdots\!64\)\( T^{24} + \)\(29\!\cdots\!62\)\( T^{26} + \)\(24\!\cdots\!13\)\( T^{28} + \)\(19\!\cdots\!51\)\( T^{30} + \)\(14\!\cdots\!76\)\( T^{32} + \)\(19\!\cdots\!51\)\( p^{2} T^{34} + \)\(24\!\cdots\!13\)\( p^{4} T^{36} + \)\(29\!\cdots\!62\)\( p^{6} T^{38} + \)\(33\!\cdots\!64\)\( p^{8} T^{40} + \)\(35\!\cdots\!00\)\( p^{10} T^{42} + \)\(34\!\cdots\!53\)\( p^{12} T^{44} + \)\(30\!\cdots\!63\)\( p^{14} T^{46} + 25052489398240412203 p^{16} T^{48} + 185788521405772448 p^{18} T^{50} + 1228366591907076 p^{20} T^{52} + 7100784349181 p^{22} T^{54} + 34928550698 p^{24} T^{56} + 140521605 p^{26} T^{58} + 434174 p^{28} T^{60} + 918 p^{30} T^{62} + p^{32} T^{64} \)
73 \( ( 1 + 30 T + 828 T^{2} + 15033 T^{3} + 269634 T^{4} + 3885947 T^{5} + 55600038 T^{6} + 679391311 T^{7} + 8317293075 T^{8} + 89942856535 T^{9} + 982571182179 T^{10} + 9640042236424 T^{11} + 96500309891964 T^{12} + 878844600373714 T^{13} + 8245980823597217 T^{14} + 70858131862084342 T^{15} + 631441189280779296 T^{16} + 70858131862084342 p T^{17} + 8245980823597217 p^{2} T^{18} + 878844600373714 p^{3} T^{19} + 96500309891964 p^{4} T^{20} + 9640042236424 p^{5} T^{21} + 982571182179 p^{6} T^{22} + 89942856535 p^{7} T^{23} + 8317293075 p^{8} T^{24} + 679391311 p^{9} T^{25} + 55600038 p^{10} T^{26} + 3885947 p^{11} T^{27} + 269634 p^{12} T^{28} + 15033 p^{13} T^{29} + 828 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
79 \( ( 1 + 37 T + 1380 T^{2} + 33837 T^{3} + 780506 T^{4} + 14779764 T^{5} + 262609683 T^{6} + 4120026414 T^{7} + 61004437274 T^{8} + 824185059612 T^{9} + 10569422541513 T^{10} + 125812949014107 T^{11} + 1427570021125438 T^{12} + 15177025392996267 T^{13} + 154214229551560808 T^{14} + 1475161164044604054 T^{15} + 13503575370193319306 T^{16} + 1475161164044604054 p T^{17} + 154214229551560808 p^{2} T^{18} + 15177025392996267 p^{3} T^{19} + 1427570021125438 p^{4} T^{20} + 125812949014107 p^{5} T^{21} + 10569422541513 p^{6} T^{22} + 824185059612 p^{7} T^{23} + 61004437274 p^{8} T^{24} + 4120026414 p^{9} T^{25} + 262609683 p^{10} T^{26} + 14779764 p^{11} T^{27} + 780506 p^{12} T^{28} + 33837 p^{13} T^{29} + 1380 p^{14} T^{30} + 37 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
83 \( 1 + 822 T^{2} + 343349 T^{4} + 95914392 T^{6} + 20064970184 T^{8} + 3356079287730 T^{10} + 471379186290416 T^{12} + 58181874982349717 T^{14} + 6606641658661670960 T^{16} + \)\(71\!\cdots\!93\)\( T^{18} + \)\(76\!\cdots\!26\)\( T^{20} + \)\(79\!\cdots\!42\)\( T^{22} + \)\(78\!\cdots\!68\)\( T^{24} + \)\(74\!\cdots\!72\)\( T^{26} + \)\(66\!\cdots\!01\)\( T^{28} + \)\(57\!\cdots\!16\)\( T^{30} + \)\(47\!\cdots\!42\)\( T^{32} + \)\(57\!\cdots\!16\)\( p^{2} T^{34} + \)\(66\!\cdots\!01\)\( p^{4} T^{36} + \)\(74\!\cdots\!72\)\( p^{6} T^{38} + \)\(78\!\cdots\!68\)\( p^{8} T^{40} + \)\(79\!\cdots\!42\)\( p^{10} T^{42} + \)\(76\!\cdots\!26\)\( p^{12} T^{44} + \)\(71\!\cdots\!93\)\( p^{14} T^{46} + 6606641658661670960 p^{16} T^{48} + 58181874982349717 p^{18} T^{50} + 471379186290416 p^{20} T^{52} + 3356079287730 p^{22} T^{54} + 20064970184 p^{24} T^{56} + 95914392 p^{26} T^{58} + 343349 p^{28} T^{60} + 822 p^{30} T^{62} + p^{32} T^{64} \)
89 \( 1 + 1488 T^{2} + 1112406 T^{4} + 557147677 T^{6} + 210347268099 T^{8} + 63854534758678 T^{10} + 16231377276976501 T^{12} + 3551572627885169738 T^{14} + \)\(68\!\cdots\!34\)\( T^{16} + \)\(11\!\cdots\!90\)\( T^{18} + \)\(17\!\cdots\!85\)\( T^{20} + \)\(25\!\cdots\!80\)\( T^{22} + \)\(32\!\cdots\!29\)\( T^{24} + \)\(37\!\cdots\!09\)\( T^{26} + \)\(41\!\cdots\!60\)\( T^{28} + \)\(41\!\cdots\!04\)\( T^{30} + \)\(37\!\cdots\!06\)\( T^{32} + \)\(41\!\cdots\!04\)\( p^{2} T^{34} + \)\(41\!\cdots\!60\)\( p^{4} T^{36} + \)\(37\!\cdots\!09\)\( p^{6} T^{38} + \)\(32\!\cdots\!29\)\( p^{8} T^{40} + \)\(25\!\cdots\!80\)\( p^{10} T^{42} + \)\(17\!\cdots\!85\)\( p^{12} T^{44} + \)\(11\!\cdots\!90\)\( p^{14} T^{46} + \)\(68\!\cdots\!34\)\( p^{16} T^{48} + 3551572627885169738 p^{18} T^{50} + 16231377276976501 p^{20} T^{52} + 63854534758678 p^{22} T^{54} + 210347268099 p^{24} T^{56} + 557147677 p^{26} T^{58} + 1112406 p^{28} T^{60} + 1488 p^{30} T^{62} + p^{32} T^{64} \)
97 \( ( 1 + 22 T + 831 T^{2} + 16657 T^{3} + 376433 T^{4} + 6520277 T^{5} + 115512375 T^{6} + 1758038966 T^{7} + 26522748021 T^{8} + 361484957147 T^{9} + 4831629595327 T^{10} + 59702674184726 T^{11} + 721442133183071 T^{12} + 8155482993387416 T^{13} + 90037778241017803 T^{14} + 935455692573641797 T^{15} + 9489931788318067828 T^{16} + 935455692573641797 p T^{17} + 90037778241017803 p^{2} T^{18} + 8155482993387416 p^{3} T^{19} + 721442133183071 p^{4} T^{20} + 59702674184726 p^{5} T^{21} + 4831629595327 p^{6} T^{22} + 361484957147 p^{7} T^{23} + 26522748021 p^{8} T^{24} + 1758038966 p^{9} T^{25} + 115512375 p^{10} T^{26} + 6520277 p^{11} T^{27} + 376433 p^{12} T^{28} + 16657 p^{13} T^{29} + 831 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.41046763309321373277220694462, −1.38714402249921382784496680960, −1.31848185598751576174362142076, −1.30113546404462336011615999019, −1.27927944899865137883141531782, −1.26226916082729608563323811632, −1.26125960064843551516439781170, −1.24810865734421170113780831839, −1.23910080025190779486702230002, −1.23109092902642923581321582119, −1.21869417167546285589605540564, −1.20377590485720792860619938233, −1.13223073178406970027683502675, −1.07980143346012740124731621700, −1.04294021971578570415275325450, −0.992384413325663856364827561683, −0.983866491224366985875385250006, −0.949460666662730038074772411714, −0.946001048035913700547777939165, −0.933043255089307877706993884493, −0.897280129206787925324532153400, −0.865063480194679232566227100448, −0.847737433780786326464855758232, −0.847466171499621743089099854139, −0.824712640035015421049780521359, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.824712640035015421049780521359, 0.847466171499621743089099854139, 0.847737433780786326464855758232, 0.865063480194679232566227100448, 0.897280129206787925324532153400, 0.933043255089307877706993884493, 0.946001048035913700547777939165, 0.949460666662730038074772411714, 0.983866491224366985875385250006, 0.992384413325663856364827561683, 1.04294021971578570415275325450, 1.07980143346012740124731621700, 1.13223073178406970027683502675, 1.20377590485720792860619938233, 1.21869417167546285589605540564, 1.23109092902642923581321582119, 1.23910080025190779486702230002, 1.24810865734421170113780831839, 1.26125960064843551516439781170, 1.26226916082729608563323811632, 1.27927944899865137883141531782, 1.30113546404462336011615999019, 1.31848185598751576174362142076, 1.38714402249921382784496680960, 1.41046763309321373277220694462

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

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