Properties

Degree 64
Conductor $ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 128·4-s − 8·5-s + 24·7-s + 704·8-s − 128·10-s + 16·11-s + 32·13-s + 384·14-s + 3.08e3·16-s − 56·17-s − 1.02e3·20-s + 256·22-s + 18·25-s + 512·26-s + 3.07e3·28-s + 112·31-s + 1.17e4·32-s − 896·34-s − 192·35-s − 52·37-s − 5.63e3·40-s − 336·41-s − 312·43-s + 2.04e3·44-s − 212·47-s + 288·49-s + ⋯
L(s)  = 1  + 8·2-s + 32·4-s − 8/5·5-s + 24/7·7-s + 88·8-s − 12.7·10-s + 1.45·11-s + 2.46·13-s + 27.4·14-s + 193·16-s − 3.29·17-s − 51.1·20-s + 11.6·22-s + 0.719·25-s + 19.6·26-s + 109.·28-s + 3.61·31-s + 366·32-s − 26.3·34-s − 5.48·35-s − 1.40·37-s − 140.·40-s − 8.19·41-s − 7.25·43-s + 46.5·44-s − 4.51·47-s + 5.87·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{210} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((64,\ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(4463.45\)
\(L(\frac12)\)  \(\approx\)  \(4463.45\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 64. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad2 \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{8} \)
3 \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \)
5 \( 1 + 8 T + 46 T^{2} + 432 T^{3} + 2876 T^{4} + 12648 T^{5} + 80236 T^{6} + 107088 p T^{7} + 448406 p T^{8} + 2798576 p T^{9} + 3595834 p^{2} T^{10} + 16140856 p^{2} T^{11} + 90810504 p^{2} T^{12} + 22593104 p^{4} T^{13} + 101467242 p^{4} T^{14} + 20798736 p^{6} T^{15} + 125615087 p^{6} T^{16} + 20798736 p^{8} T^{17} + 101467242 p^{8} T^{18} + 22593104 p^{10} T^{19} + 90810504 p^{10} T^{20} + 16140856 p^{12} T^{21} + 3595834 p^{14} T^{22} + 2798576 p^{15} T^{23} + 448406 p^{17} T^{24} + 107088 p^{19} T^{25} + 80236 p^{20} T^{26} + 12648 p^{22} T^{27} + 2876 p^{24} T^{28} + 432 p^{26} T^{29} + 46 p^{28} T^{30} + 8 p^{30} T^{31} + p^{32} T^{32} \)
7 \( 1 - 24 T + 288 T^{2} - 2124 T^{3} + 10412 T^{4} - 20700 T^{5} - 246168 T^{6} + 3874044 T^{7} - 18630734 T^{8} - 17790732 p T^{9} + 59252472 p^{2} T^{10} - 538557984 p^{2} T^{11} + 502495724 p^{3} T^{12} - 355480248 p^{4} T^{13} + 900629640 p^{4} T^{14} + 7345867716 p^{4} T^{15} - 95737559741 p^{4} T^{16} + 7345867716 p^{6} T^{17} + 900629640 p^{8} T^{18} - 355480248 p^{10} T^{19} + 502495724 p^{11} T^{20} - 538557984 p^{12} T^{21} + 59252472 p^{14} T^{22} - 17790732 p^{15} T^{23} - 18630734 p^{16} T^{24} + 3874044 p^{18} T^{25} - 246168 p^{20} T^{26} - 20700 p^{22} T^{27} + 10412 p^{24} T^{28} - 2124 p^{26} T^{29} + 288 p^{28} T^{30} - 24 p^{30} T^{31} + p^{32} T^{32} \)
good11 \( ( 1 - 8 T - 394 T^{2} + 2632 T^{3} + 8738 p T^{4} - 435460 T^{5} - 14471012 T^{6} + 2529756 T^{7} + 1392197060 T^{8} + 1251683464 p T^{9} - 21348409028 T^{10} - 3485642337764 T^{11} - 17897304342860 T^{12} + 439422412193612 T^{13} + 4201057792264490 T^{14} - 23321035185362592 T^{15} - 586451967345812685 T^{16} - 23321035185362592 p^{2} T^{17} + 4201057792264490 p^{4} T^{18} + 439422412193612 p^{6} T^{19} - 17897304342860 p^{8} T^{20} - 3485642337764 p^{10} T^{21} - 21348409028 p^{12} T^{22} + 1251683464 p^{15} T^{23} + 1392197060 p^{16} T^{24} + 2529756 p^{18} T^{25} - 14471012 p^{20} T^{26} - 435460 p^{22} T^{27} + 8738 p^{25} T^{28} + 2632 p^{26} T^{29} - 394 p^{28} T^{30} - 8 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
13 \( ( 1 - 16 T + 128 T^{2} + 300 T^{3} - 76916 T^{4} + 1210124 T^{5} - 9471736 T^{6} + 10852940 T^{7} + 3579682674 T^{8} - 50513851868 T^{9} + 356950131128 T^{10} + 634421851536 T^{11} - 115946311393164 T^{12} + 1279294176331112 T^{13} - 5145398194358520 T^{14} - 95283839662101204 T^{15} + 3399141505373101635 T^{16} - 95283839662101204 p^{2} T^{17} - 5145398194358520 p^{4} T^{18} + 1279294176331112 p^{6} T^{19} - 115946311393164 p^{8} T^{20} + 634421851536 p^{10} T^{21} + 356950131128 p^{12} T^{22} - 50513851868 p^{14} T^{23} + 3579682674 p^{16} T^{24} + 10852940 p^{18} T^{25} - 9471736 p^{20} T^{26} + 1210124 p^{22} T^{27} - 76916 p^{24} T^{28} + 300 p^{26} T^{29} + 128 p^{28} T^{30} - 16 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
17 \( 1 + 56 T + 1568 T^{2} + 12472 T^{3} - 339024 T^{4} - 10826372 T^{5} + 3088192 T^{6} + 4365845152 T^{7} + 92862207896 T^{8} + 517927361776 T^{9} - 2501052439928 T^{10} - 1764197734956 p T^{11} + 1132940821805768 T^{12} - 4015839643155236 T^{13} + 412944038970032192 T^{14} + 25313422169102668480 T^{15} + \)\(64\!\cdots\!46\)\( T^{16} + \)\(56\!\cdots\!92\)\( T^{17} + \)\(25\!\cdots\!00\)\( p T^{18} + \)\(77\!\cdots\!32\)\( T^{19} + \)\(22\!\cdots\!16\)\( T^{20} + \)\(78\!\cdots\!76\)\( p T^{21} + \)\(69\!\cdots\!60\)\( T^{22} + \)\(14\!\cdots\!36\)\( p T^{23} + \)\(44\!\cdots\!04\)\( T^{24} + \)\(53\!\cdots\!40\)\( T^{25} + \)\(15\!\cdots\!52\)\( T^{26} + \)\(17\!\cdots\!44\)\( T^{27} + \)\(55\!\cdots\!04\)\( T^{28} + \)\(43\!\cdots\!52\)\( T^{29} + \)\(16\!\cdots\!04\)\( T^{30} + \)\(72\!\cdots\!20\)\( T^{31} + \)\(30\!\cdots\!79\)\( T^{32} + \)\(72\!\cdots\!20\)\( p^{2} T^{33} + \)\(16\!\cdots\!04\)\( p^{4} T^{34} + \)\(43\!\cdots\!52\)\( p^{6} T^{35} + \)\(55\!\cdots\!04\)\( p^{8} T^{36} + \)\(17\!\cdots\!44\)\( p^{10} T^{37} + \)\(15\!\cdots\!52\)\( p^{12} T^{38} + \)\(53\!\cdots\!40\)\( p^{14} T^{39} + \)\(44\!\cdots\!04\)\( p^{16} T^{40} + \)\(14\!\cdots\!36\)\( p^{19} T^{41} + \)\(69\!\cdots\!60\)\( p^{20} T^{42} + \)\(78\!\cdots\!76\)\( p^{23} T^{43} + \)\(22\!\cdots\!16\)\( p^{24} T^{44} + \)\(77\!\cdots\!32\)\( p^{26} T^{45} + \)\(25\!\cdots\!00\)\( p^{29} T^{46} + \)\(56\!\cdots\!92\)\( p^{30} T^{47} + \)\(64\!\cdots\!46\)\( p^{32} T^{48} + 25313422169102668480 p^{34} T^{49} + 412944038970032192 p^{36} T^{50} - 4015839643155236 p^{38} T^{51} + 1132940821805768 p^{40} T^{52} - 1764197734956 p^{43} T^{53} - 2501052439928 p^{44} T^{54} + 517927361776 p^{46} T^{55} + 92862207896 p^{48} T^{56} + 4365845152 p^{50} T^{57} + 3088192 p^{52} T^{58} - 10826372 p^{54} T^{59} - 339024 p^{56} T^{60} + 12472 p^{58} T^{61} + 1568 p^{60} T^{62} + 56 p^{62} T^{63} + p^{64} T^{64} \)
19 \( 1 + 2776 T^{2} + 3881980 T^{4} + 3576281552 T^{6} + 2386586138342 T^{8} + 1187403698603024 T^{10} + 419969200446909720 T^{12} + 72475725935958717456 T^{14} - \)\(30\!\cdots\!07\)\( T^{16} - \)\(37\!\cdots\!92\)\( T^{18} - \)\(21\!\cdots\!24\)\( T^{20} - \)\(83\!\cdots\!20\)\( T^{22} - \)\(22\!\cdots\!06\)\( T^{24} - \)\(16\!\cdots\!68\)\( T^{26} + \)\(22\!\cdots\!12\)\( T^{28} + \)\(16\!\cdots\!16\)\( T^{30} + \)\(71\!\cdots\!92\)\( T^{32} + \)\(16\!\cdots\!16\)\( p^{4} T^{34} + \)\(22\!\cdots\!12\)\( p^{8} T^{36} - \)\(16\!\cdots\!68\)\( p^{12} T^{38} - \)\(22\!\cdots\!06\)\( p^{16} T^{40} - \)\(83\!\cdots\!20\)\( p^{20} T^{42} - \)\(21\!\cdots\!24\)\( p^{24} T^{44} - \)\(37\!\cdots\!92\)\( p^{28} T^{46} - \)\(30\!\cdots\!07\)\( p^{32} T^{48} + 72475725935958717456 p^{36} T^{50} + 419969200446909720 p^{40} T^{52} + 1187403698603024 p^{44} T^{54} + 2386586138342 p^{48} T^{56} + 3576281552 p^{52} T^{58} + 3881980 p^{56} T^{60} + 2776 p^{60} T^{62} + p^{64} T^{64} \)
23 \( 1 + 68024 T^{3} - 134544 T^{4} + 22408476 T^{5} + 2313632288 T^{6} - 5304181952 T^{7} + 1518854648968 T^{8} + 47861612276592 T^{9} + 201543567775112 T^{10} + 51005637111202356 T^{11} + 661867032456789352 T^{12} + 19061067729651745868 T^{13} + \)\(10\!\cdots\!32\)\( T^{14} + \)\(91\!\cdots\!24\)\( T^{15} + \)\(65\!\cdots\!02\)\( T^{16} + \)\(15\!\cdots\!20\)\( T^{17} + \)\(23\!\cdots\!80\)\( T^{18} + \)\(13\!\cdots\!96\)\( T^{19} + \)\(17\!\cdots\!68\)\( T^{20} + \)\(64\!\cdots\!12\)\( T^{21} + \)\(19\!\cdots\!44\)\( T^{22} + \)\(24\!\cdots\!12\)\( T^{23} + \)\(56\!\cdots\!08\)\( p T^{24} + \)\(20\!\cdots\!60\)\( T^{25} + \)\(50\!\cdots\!28\)\( T^{26} + \)\(18\!\cdots\!28\)\( T^{27} + \)\(20\!\cdots\!24\)\( T^{28} + \)\(96\!\cdots\!08\)\( T^{29} + \)\(19\!\cdots\!44\)\( T^{30} + \)\(30\!\cdots\!48\)\( T^{31} + \)\(13\!\cdots\!55\)\( T^{32} + \)\(30\!\cdots\!48\)\( p^{2} T^{33} + \)\(19\!\cdots\!44\)\( p^{4} T^{34} + \)\(96\!\cdots\!08\)\( p^{6} T^{35} + \)\(20\!\cdots\!24\)\( p^{8} T^{36} + \)\(18\!\cdots\!28\)\( p^{10} T^{37} + \)\(50\!\cdots\!28\)\( p^{12} T^{38} + \)\(20\!\cdots\!60\)\( p^{14} T^{39} + \)\(56\!\cdots\!08\)\( p^{17} T^{40} + \)\(24\!\cdots\!12\)\( p^{18} T^{41} + \)\(19\!\cdots\!44\)\( p^{20} T^{42} + \)\(64\!\cdots\!12\)\( p^{22} T^{43} + \)\(17\!\cdots\!68\)\( p^{24} T^{44} + \)\(13\!\cdots\!96\)\( p^{26} T^{45} + \)\(23\!\cdots\!80\)\( p^{28} T^{46} + \)\(15\!\cdots\!20\)\( p^{30} T^{47} + \)\(65\!\cdots\!02\)\( p^{32} T^{48} + \)\(91\!\cdots\!24\)\( p^{34} T^{49} + \)\(10\!\cdots\!32\)\( p^{36} T^{50} + 19061067729651745868 p^{38} T^{51} + 661867032456789352 p^{40} T^{52} + 51005637111202356 p^{42} T^{53} + 201543567775112 p^{44} T^{54} + 47861612276592 p^{46} T^{55} + 1518854648968 p^{48} T^{56} - 5304181952 p^{50} T^{57} + 2313632288 p^{52} T^{58} + 22408476 p^{54} T^{59} - 134544 p^{56} T^{60} + 68024 p^{58} T^{61} + p^{64} T^{64} \)
29 \( ( 1 - 6528 T^{2} + 22826508 T^{4} - 54745178940 T^{6} + 100071828912532 T^{8} - 147349080649060692 T^{10} + \)\(18\!\cdots\!64\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!10\)\( T^{16} - \)\(18\!\cdots\!00\)\( p^{4} T^{18} + \)\(18\!\cdots\!64\)\( p^{8} T^{20} - 147349080649060692 p^{12} T^{22} + 100071828912532 p^{16} T^{24} - 54745178940 p^{20} T^{26} + 22826508 p^{24} T^{28} - 6528 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
31 \( ( 1 - 56 T - 3388 T^{2} + 239664 T^{3} + 6678350 T^{4} - 596615600 T^{5} - 8002232664 T^{6} + 1047432699472 T^{7} + 3843579706249 T^{8} - 1395589893400072 T^{9} + 7151053393405784 T^{10} + 1433793427756674072 T^{11} - 22161583062727752498 T^{12} - \)\(10\!\cdots\!36\)\( T^{13} + \)\(34\!\cdots\!00\)\( T^{14} + \)\(38\!\cdots\!36\)\( T^{15} - \)\(38\!\cdots\!48\)\( T^{16} + \)\(38\!\cdots\!36\)\( p^{2} T^{17} + \)\(34\!\cdots\!00\)\( p^{4} T^{18} - \)\(10\!\cdots\!36\)\( p^{6} T^{19} - 22161583062727752498 p^{8} T^{20} + 1433793427756674072 p^{10} T^{21} + 7151053393405784 p^{12} T^{22} - 1395589893400072 p^{14} T^{23} + 3843579706249 p^{16} T^{24} + 1047432699472 p^{18} T^{25} - 8002232664 p^{20} T^{26} - 596615600 p^{22} T^{27} + 6678350 p^{24} T^{28} + 239664 p^{26} T^{29} - 3388 p^{28} T^{30} - 56 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
37 \( 1 + 52 T + 1352 T^{2} + 122600 T^{3} + 3182200 T^{4} + 166230120 T^{5} + 11857011840 T^{6} + 266793416960 T^{7} + 22301586254862 T^{8} + 609687320822112 T^{9} + 10345712305831600 T^{10} + 1572083937547540596 T^{11} + 19547263194413223256 T^{12} + \)\(15\!\cdots\!16\)\( T^{13} + \)\(84\!\cdots\!80\)\( T^{14} + \)\(18\!\cdots\!28\)\( T^{15} + \)\(15\!\cdots\!73\)\( T^{16} + \)\(26\!\cdots\!88\)\( T^{17} + \)\(83\!\cdots\!92\)\( T^{18} + \)\(11\!\cdots\!76\)\( T^{19} + \)\(34\!\cdots\!88\)\( T^{20} + \)\(11\!\cdots\!60\)\( T^{21} + \)\(13\!\cdots\!92\)\( p T^{22} + \)\(40\!\cdots\!64\)\( T^{23} + \)\(10\!\cdots\!42\)\( T^{24} + \)\(10\!\cdots\!60\)\( T^{25} + \)\(60\!\cdots\!64\)\( T^{26} + \)\(53\!\cdots\!12\)\( T^{27} - \)\(78\!\cdots\!00\)\( T^{28} + \)\(66\!\cdots\!20\)\( T^{29} + \)\(16\!\cdots\!76\)\( T^{30} + \)\(43\!\cdots\!48\)\( T^{31} + \)\(38\!\cdots\!92\)\( T^{32} + \)\(43\!\cdots\!48\)\( p^{2} T^{33} + \)\(16\!\cdots\!76\)\( p^{4} T^{34} + \)\(66\!\cdots\!20\)\( p^{6} T^{35} - \)\(78\!\cdots\!00\)\( p^{8} T^{36} + \)\(53\!\cdots\!12\)\( p^{10} T^{37} + \)\(60\!\cdots\!64\)\( p^{12} T^{38} + \)\(10\!\cdots\!60\)\( p^{14} T^{39} + \)\(10\!\cdots\!42\)\( p^{16} T^{40} + \)\(40\!\cdots\!64\)\( p^{18} T^{41} + \)\(13\!\cdots\!92\)\( p^{21} T^{42} + \)\(11\!\cdots\!60\)\( p^{22} T^{43} + \)\(34\!\cdots\!88\)\( p^{24} T^{44} + \)\(11\!\cdots\!76\)\( p^{26} T^{45} + \)\(83\!\cdots\!92\)\( p^{28} T^{46} + \)\(26\!\cdots\!88\)\( p^{30} T^{47} + \)\(15\!\cdots\!73\)\( p^{32} T^{48} + \)\(18\!\cdots\!28\)\( p^{34} T^{49} + \)\(84\!\cdots\!80\)\( p^{36} T^{50} + \)\(15\!\cdots\!16\)\( p^{38} T^{51} + 19547263194413223256 p^{40} T^{52} + 1572083937547540596 p^{42} T^{53} + 10345712305831600 p^{44} T^{54} + 609687320822112 p^{46} T^{55} + 22301586254862 p^{48} T^{56} + 266793416960 p^{50} T^{57} + 11857011840 p^{52} T^{58} + 166230120 p^{54} T^{59} + 3182200 p^{56} T^{60} + 122600 p^{58} T^{61} + 1352 p^{60} T^{62} + 52 p^{62} T^{63} + p^{64} T^{64} \)
41 \( ( 1 + 84 T + 12254 T^{2} + 755636 T^{3} + 63094622 T^{4} + 3129227332 T^{5} + 192188697748 T^{6} + 7927642087344 T^{7} + 390034191082528 T^{8} + 7927642087344 p^{2} T^{9} + 192188697748 p^{4} T^{10} + 3129227332 p^{6} T^{11} + 63094622 p^{8} T^{12} + 755636 p^{10} T^{13} + 12254 p^{12} T^{14} + 84 p^{14} T^{15} + p^{16} T^{16} )^{4} \)
43 \( ( 1 + 156 T + 12168 T^{2} + 779364 T^{3} + 53542704 T^{4} + 3488424888 T^{5} + 196390782504 T^{6} + 10519123270152 T^{7} + 570621824255746 T^{8} + 29729215438744788 T^{9} + 1461917802887868384 T^{10} + 71587117180546971492 T^{11} + \)\(34\!\cdots\!52\)\( T^{12} + \)\(16\!\cdots\!24\)\( T^{13} + \)\(71\!\cdots\!72\)\( T^{14} + \)\(32\!\cdots\!96\)\( T^{15} + \)\(14\!\cdots\!03\)\( T^{16} + \)\(32\!\cdots\!96\)\( p^{2} T^{17} + \)\(71\!\cdots\!72\)\( p^{4} T^{18} + \)\(16\!\cdots\!24\)\( p^{6} T^{19} + \)\(34\!\cdots\!52\)\( p^{8} T^{20} + 71587117180546971492 p^{10} T^{21} + 1461917802887868384 p^{12} T^{22} + 29729215438744788 p^{14} T^{23} + 570621824255746 p^{16} T^{24} + 10519123270152 p^{18} T^{25} + 196390782504 p^{20} T^{26} + 3488424888 p^{22} T^{27} + 53542704 p^{24} T^{28} + 779364 p^{26} T^{29} + 12168 p^{28} T^{30} + 156 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
47 \( 1 + 212 T + 22472 T^{2} + 1307368 T^{3} + 27251844 T^{4} - 2262609212 T^{5} - 237471047600 T^{6} - 9795487606532 T^{7} - 96899309821876 T^{8} + 10028123617033048 T^{9} + 528400622777080744 T^{10} + 14236505108278001916 T^{11} + \)\(67\!\cdots\!12\)\( T^{12} + \)\(41\!\cdots\!12\)\( T^{13} + \)\(24\!\cdots\!68\)\( T^{14} - \)\(14\!\cdots\!60\)\( T^{15} - \)\(98\!\cdots\!14\)\( T^{16} - \)\(17\!\cdots\!00\)\( T^{17} + \)\(94\!\cdots\!76\)\( T^{18} + \)\(13\!\cdots\!08\)\( p T^{19} + \)\(15\!\cdots\!64\)\( T^{20} + \)\(64\!\cdots\!80\)\( T^{21} + \)\(44\!\cdots\!40\)\( T^{22} - \)\(15\!\cdots\!60\)\( T^{23} - \)\(20\!\cdots\!68\)\( T^{24} - \)\(97\!\cdots\!68\)\( T^{25} + \)\(21\!\cdots\!36\)\( T^{26} + \)\(40\!\cdots\!08\)\( T^{27} + \)\(15\!\cdots\!80\)\( T^{28} - \)\(95\!\cdots\!32\)\( T^{29} - \)\(27\!\cdots\!28\)\( T^{30} - \)\(86\!\cdots\!56\)\( T^{31} - \)\(17\!\cdots\!61\)\( T^{32} - \)\(86\!\cdots\!56\)\( p^{2} T^{33} - \)\(27\!\cdots\!28\)\( p^{4} T^{34} - \)\(95\!\cdots\!32\)\( p^{6} T^{35} + \)\(15\!\cdots\!80\)\( p^{8} T^{36} + \)\(40\!\cdots\!08\)\( p^{10} T^{37} + \)\(21\!\cdots\!36\)\( p^{12} T^{38} - \)\(97\!\cdots\!68\)\( p^{14} T^{39} - \)\(20\!\cdots\!68\)\( p^{16} T^{40} - \)\(15\!\cdots\!60\)\( p^{18} T^{41} + \)\(44\!\cdots\!40\)\( p^{20} T^{42} + \)\(64\!\cdots\!80\)\( p^{22} T^{43} + \)\(15\!\cdots\!64\)\( p^{24} T^{44} + \)\(13\!\cdots\!08\)\( p^{27} T^{45} + \)\(94\!\cdots\!76\)\( p^{28} T^{46} - \)\(17\!\cdots\!00\)\( p^{30} T^{47} - \)\(98\!\cdots\!14\)\( p^{32} T^{48} - \)\(14\!\cdots\!60\)\( p^{34} T^{49} + \)\(24\!\cdots\!68\)\( p^{36} T^{50} + \)\(41\!\cdots\!12\)\( p^{38} T^{51} + \)\(67\!\cdots\!12\)\( p^{40} T^{52} + 14236505108278001916 p^{42} T^{53} + 528400622777080744 p^{44} T^{54} + 10028123617033048 p^{46} T^{55} - 96899309821876 p^{48} T^{56} - 9795487606532 p^{50} T^{57} - 237471047600 p^{52} T^{58} - 2262609212 p^{54} T^{59} + 27251844 p^{56} T^{60} + 1307368 p^{58} T^{61} + 22472 p^{60} T^{62} + 212 p^{62} T^{63} + p^{64} T^{64} \)
53 \( 1 + 96 T + 4608 T^{2} + 630784 T^{3} + 41183576 T^{4} + 1688464384 T^{5} + 171262889984 T^{6} + 9807168345376 T^{7} + 483509317295076 T^{8} + 45461102504219072 T^{9} + 2694697526261522432 T^{10} + \)\(14\!\cdots\!64\)\( T^{11} + \)\(19\!\cdots\!76\)\( p T^{12} + \)\(56\!\cdots\!64\)\( T^{13} + \)\(29\!\cdots\!16\)\( T^{14} + \)\(18\!\cdots\!76\)\( T^{15} + \)\(10\!\cdots\!70\)\( T^{16} + \)\(59\!\cdots\!80\)\( T^{17} + \)\(36\!\cdots\!36\)\( T^{18} + \)\(21\!\cdots\!92\)\( T^{19} + \)\(12\!\cdots\!04\)\( T^{20} + \)\(62\!\cdots\!48\)\( T^{21} + \)\(35\!\cdots\!28\)\( T^{22} + \)\(18\!\cdots\!00\)\( T^{23} + \)\(95\!\cdots\!08\)\( T^{24} + \)\(56\!\cdots\!76\)\( T^{25} + \)\(32\!\cdots\!40\)\( T^{26} + \)\(17\!\cdots\!96\)\( T^{27} + \)\(97\!\cdots\!52\)\( T^{28} + \)\(52\!\cdots\!20\)\( T^{29} + \)\(25\!\cdots\!08\)\( T^{30} + \)\(13\!\cdots\!72\)\( T^{31} + \)\(71\!\cdots\!35\)\( T^{32} + \)\(13\!\cdots\!72\)\( p^{2} T^{33} + \)\(25\!\cdots\!08\)\( p^{4} T^{34} + \)\(52\!\cdots\!20\)\( p^{6} T^{35} + \)\(97\!\cdots\!52\)\( p^{8} T^{36} + \)\(17\!\cdots\!96\)\( p^{10} T^{37} + \)\(32\!\cdots\!40\)\( p^{12} T^{38} + \)\(56\!\cdots\!76\)\( p^{14} T^{39} + \)\(95\!\cdots\!08\)\( p^{16} T^{40} + \)\(18\!\cdots\!00\)\( p^{18} T^{41} + \)\(35\!\cdots\!28\)\( p^{20} T^{42} + \)\(62\!\cdots\!48\)\( p^{22} T^{43} + \)\(12\!\cdots\!04\)\( p^{24} T^{44} + \)\(21\!\cdots\!92\)\( p^{26} T^{45} + \)\(36\!\cdots\!36\)\( p^{28} T^{46} + \)\(59\!\cdots\!80\)\( p^{30} T^{47} + \)\(10\!\cdots\!70\)\( p^{32} T^{48} + \)\(18\!\cdots\!76\)\( p^{34} T^{49} + \)\(29\!\cdots\!16\)\( p^{36} T^{50} + \)\(56\!\cdots\!64\)\( p^{38} T^{51} + \)\(19\!\cdots\!76\)\( p^{41} T^{52} + \)\(14\!\cdots\!64\)\( p^{42} T^{53} + 2694697526261522432 p^{44} T^{54} + 45461102504219072 p^{46} T^{55} + 483509317295076 p^{48} T^{56} + 9807168345376 p^{50} T^{57} + 171262889984 p^{52} T^{58} + 1688464384 p^{54} T^{59} + 41183576 p^{56} T^{60} + 630784 p^{58} T^{61} + 4608 p^{60} T^{62} + 96 p^{62} T^{63} + p^{64} T^{64} \)
59 \( 1 + 34792 T^{2} + 633178676 T^{4} + 8016858213800 T^{6} + 78908075606024860 T^{8} + \)\(63\!\cdots\!92\)\( T^{10} + \)\(43\!\cdots\!04\)\( T^{12} + \)\(25\!\cdots\!32\)\( T^{14} + \)\(13\!\cdots\!10\)\( T^{16} + \)\(60\!\cdots\!80\)\( T^{18} + \)\(23\!\cdots\!56\)\( T^{20} + \)\(78\!\cdots\!92\)\( T^{22} + \)\(20\!\cdots\!96\)\( T^{24} + \)\(34\!\cdots\!80\)\( T^{26} - \)\(26\!\cdots\!20\)\( T^{28} - \)\(48\!\cdots\!52\)\( T^{30} - \)\(22\!\cdots\!29\)\( T^{32} - \)\(48\!\cdots\!52\)\( p^{4} T^{34} - \)\(26\!\cdots\!20\)\( p^{8} T^{36} + \)\(34\!\cdots\!80\)\( p^{12} T^{38} + \)\(20\!\cdots\!96\)\( p^{16} T^{40} + \)\(78\!\cdots\!92\)\( p^{20} T^{42} + \)\(23\!\cdots\!56\)\( p^{24} T^{44} + \)\(60\!\cdots\!80\)\( p^{28} T^{46} + \)\(13\!\cdots\!10\)\( p^{32} T^{48} + \)\(25\!\cdots\!32\)\( p^{36} T^{50} + \)\(43\!\cdots\!04\)\( p^{40} T^{52} + \)\(63\!\cdots\!92\)\( p^{44} T^{54} + 78908075606024860 p^{48} T^{56} + 8016858213800 p^{52} T^{58} + 633178676 p^{56} T^{60} + 34792 p^{60} T^{62} + p^{64} T^{64} \)
61 \( ( 1 - 108 T - 9776 T^{2} + 1287280 T^{3} + 64209932 T^{4} - 9271608512 T^{5} - 245281248808 T^{6} + 43276283373420 T^{7} + 442293509205086 T^{8} - 146363882986939216 T^{9} + 2220709596061860020 T^{10} + \)\(34\!\cdots\!08\)\( T^{11} - \)\(24\!\cdots\!68\)\( T^{12} - \)\(61\!\cdots\!52\)\( T^{13} + \)\(14\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!80\)\( T^{15} - \)\(61\!\cdots\!09\)\( T^{16} + \)\(49\!\cdots\!80\)\( p^{2} T^{17} + \)\(14\!\cdots\!60\)\( p^{4} T^{18} - \)\(61\!\cdots\!52\)\( p^{6} T^{19} - \)\(24\!\cdots\!68\)\( p^{8} T^{20} + \)\(34\!\cdots\!08\)\( p^{10} T^{21} + 2220709596061860020 p^{12} T^{22} - 146363882986939216 p^{14} T^{23} + 442293509205086 p^{16} T^{24} + 43276283373420 p^{18} T^{25} - 245281248808 p^{20} T^{26} - 9271608512 p^{22} T^{27} + 64209932 p^{24} T^{28} + 1287280 p^{26} T^{29} - 9776 p^{28} T^{30} - 108 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
67 \( 1 - 128T + 8.19e3T^{2} - 1.08e6T^{3} + 6.75e7T^{4} - 5.23e8T^{5} + 9.85e10T^{6} + 5.36e12T^{7} - 1.54e15T^{8} + 3.65e15T^{9} - 1.01e17T^{10} + 1.11e20T^{11} + 5.25e22T^{12} - 4.41e24T^{13} + 1.86e26T^{14} - 3.02e28T^{15} + 1.77e30T^{16} + 7.67e28T^{17} + 3.23e33T^{18} + 1.14e34T^{19} - 3.48e37T^{20} - 2.52e38T^{21} + 3.38e40T^{22} + 2.47e42T^{23} + 9.27e44T^{24} - 7.05e46T^{25} + 1.87e48T^{26} - 3.65e50T^{27} + 2.16e52T^{28} + 1.91e53T^{29} + 3.33e55T^{30} - 7.77e56T^{31}+O(T^{32}) \)
71 \( 1 + 848T + 4.00e5T^{2} + 1.34e8T^{3} + 3.55e10T^{4} + 7.77e12T^{5} + 1.45e15T^{6} + 2.39e17T^{7} + 3.51e19T^{8} + 4.66e21T^{9} + 5.64e23T^{10} + 6.29e25T^{11} + 6.54e27T^{12} + 6.39e29T^{13} + 5.95e31T^{14} + 5.35e33T^{15} + 4.70e35T^{16} + 4.08e37T^{17} + 3.53e39T^{18} + 3.04e41T^{19} + 2.59e43T^{20} + 2.17e45T^{21} + 1.78e47T^{22} + 1.42e49T^{23} + 1.11e51T^{24} + 8.56e52T^{25} + 6.49e54T^{26} + 4.89e56T^{27} + 3.67e58T^{28} + 2.74e60T^{29} + 2.03e62T^{30} + 1.48e64T^{31}+O(T^{32}) \)
73 \( 1 - 84T + 3.52e3T^{2} + 5.65e5T^{3} - 1.07e8T^{4} + 7.85e9T^{5} - 1.19e11T^{6} - 4.23e13T^{7} + 6.57e15T^{8} - 4.21e17T^{9} + 6.24e18T^{10} + 2.26e21T^{11} - 3.32e23T^{12} + 2.09e25T^{13} - 2.64e26T^{14} - 1.11e29T^{15} + 1.49e31T^{16} - 8.99e32T^{17} + 7.83e33T^{18} + 4.79e36T^{19} - 5.91e38T^{20} + 3.42e40T^{21} - 2.35e41T^{22} - 1.81e44T^{23} + 2.14e46T^{24} - 1.18e48T^{25} + 4.31e48T^{26} + 6.27e51T^{27} - 6.95e53T^{28} + 3.60e55T^{29} - 3.85e55T^{30} - 1.93e59T^{31}+O(T^{32}) \)
79 \( 1 + 5.11e4T^{2} + 1.24e9T^{4} + 1.95e13T^{6} + 2.34e17T^{8} + 2.39e21T^{10} + 2.21e25T^{12} + 1.89e29T^{14} + 1.52e33T^{16} + 1.18e37T^{18} + 8.81e40T^{20} + 6.36e44T^{22} + 4.45e48T^{24} + 3.02e52T^{26} + 1.99e56T^{28} + 1.28e60T^{30}+O(T^{31}) \)
83 \( 1 + 416T + 8.65e4T^{2} + 1.31e7T^{3} + 1.45e9T^{4} + 9.89e10T^{5} + 1.54e12T^{6} - 6.63e14T^{7} - 1.01e17T^{8} - 6.46e18T^{9} + 6.59e19T^{10} + 6.32e22T^{11} + 6.98e24T^{12} + 2.58e26T^{13} - 2.29e28T^{14} - 4.26e30T^{15} - 2.09e32T^{16} + 1.99e34T^{17} + 4.32e36T^{18} + 3.70e38T^{19} + 8.08e39T^{20} - 1.82e42T^{21} - 2.33e44T^{22} - 1.06e46T^{23} + 7.58e47T^{24} + 1.54e50T^{25} + 9.90e51T^{26} - 1.10e53T^{27} - 8.77e55T^{28} - 7.49e57T^{29} - 9.26e58T^{30}+O(T^{31}) \)
89 \( 1 + 1.49e4T^{2} - 2.50e8T^{4} - 4.51e12T^{6} + 3.90e16T^{8} + 7.98e20T^{10} - 4.28e24T^{12} - 9.96e28T^{14} + 3.50e32T^{16} + 9.39e36T^{18} - 2.12e40T^{20} - 6.71e44T^{22} + 9.25e47T^{24} + 3.42e52T^{26} - 2.56e55T^{28}+O(T^{30}) \)
97 \( 1 - 488T + 1.19e5T^{2} - 1.75e7T^{3} + 1.05e9T^{4} + 1.60e11T^{5} - 4.92e13T^{6} + 5.62e15T^{7} - 3.11e16T^{8} - 9.81e19T^{9} + 1.75e22T^{10} - 1.40e24T^{11} - 3.74e25T^{12} + 2.58e28T^{13} - 3.41e30T^{14} + 1.59e32T^{15} + 2.32e34T^{16} - 5.64e36T^{17} + 5.45e38T^{18} - 1.04e40T^{19} - 5.11e42T^{20} + 8.37e44T^{21} - 5.72e46T^{22} - 1.65e48T^{23} + 8.87e50T^{24} - 1.03e53T^{25} + 4.64e54T^{26} + 4.56e56T^{27} - 1.08e59T^{28} + 9.39e60T^{29}+O(T^{30}) \)
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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

−2.10792401412545052848177456450, −2.05138926077198614200135509175, −1.96601886466453838492462458119, −1.84093189062143835193952411122, −1.82803506290021865480101859633, −1.69037018586905380950111495725, −1.58274061973270565968604083577, −1.52150681072174324212917559406, −1.51519254987444798371562829155, −1.49301966346859662059372150149, −1.49092148665303647052783058366, −1.45379186528364725593391675206, −1.43035623140440470050321445640, −1.41274170846636405237494959365, −1.35965905608254561319358197464, −1.15314953742729619255252851704, −1.11062738048683047225672954823, −0.892446552804224083357942211937, −0.833836080486318294907237642833, −0.65554813988911445324829791987, −0.52507656120272725444614665177, −0.37600600752498121990977456478, −0.23404371264569285086575073358, −0.22289180705527761589741784040, −0.19685344721207137413332398144, 0.19685344721207137413332398144, 0.22289180705527761589741784040, 0.23404371264569285086575073358, 0.37600600752498121990977456478, 0.52507656120272725444614665177, 0.65554813988911445324829791987, 0.833836080486318294907237642833, 0.892446552804224083357942211937, 1.11062738048683047225672954823, 1.15314953742729619255252851704, 1.35965905608254561319358197464, 1.41274170846636405237494959365, 1.43035623140440470050321445640, 1.45379186528364725593391675206, 1.49092148665303647052783058366, 1.49301966346859662059372150149, 1.51519254987444798371562829155, 1.52150681072174324212917559406, 1.58274061973270565968604083577, 1.69037018586905380950111495725, 1.82803506290021865480101859633, 1.84093189062143835193952411122, 1.96601886466453838492462458119, 2.05138926077198614200135509175, 2.10792401412545052848177456450

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

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