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 − 8·7-s − 704·8-s − 32·11-s − 32·13-s + 128·14-s + 3.08e3·16-s − 56·17-s + 512·22-s − 48·23-s + 34·25-s + 512·26-s − 1.02e3·28-s + 160·31-s − 1.17e4·32-s + 896·34-s + 44·37-s − 80·41-s − 184·43-s − 4.09e3·44-s + 768·46-s − 228·47-s + 32·49-s − 544·50-s − 4.09e3·52-s + 48·53-s + ⋯
L(s)  = 1  − 8·2-s + 32·4-s − 8/7·7-s − 88·8-s − 2.90·11-s − 2.46·13-s + 64/7·14-s + 193·16-s − 3.29·17-s + 23.2·22-s − 2.08·23-s + 1.35·25-s + 19.6·26-s − 36.5·28-s + 5.16·31-s − 366·32-s + 26.3·34-s + 1.18·37-s − 1.95·41-s − 4.27·43-s − 93.0·44-s + 16.6·46-s − 4.85·47-s + 0.653·49-s − 10.8·50-s − 78.7·52-s + 0.905·53-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  =  \((64,\ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.119837\)
\(L(\frac12)\)  \(\approx\)  \(0.119837\)
\(L(2)\)   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 \( 1 - 34 T^{2} - 336 T^{3} - 676 T^{4} + 10752 T^{5} + 87436 T^{6} + 170856 T^{7} - 1241746 T^{8} - 2894304 p T^{9} - 1752182 p^{2} T^{10} + 3560592 p^{2} T^{11} + 11472552 p^{3} T^{12} + 12160008 p^{4} T^{13} + 11630394 p^{4} T^{14} - 40712448 p^{5} T^{15} - 57899153 p^{6} T^{16} - 40712448 p^{7} T^{17} + 11630394 p^{8} T^{18} + 12160008 p^{10} T^{19} + 11472552 p^{11} T^{20} + 3560592 p^{12} T^{21} - 1752182 p^{14} T^{22} - 2894304 p^{15} T^{23} - 1241746 p^{16} T^{24} + 170856 p^{18} T^{25} + 87436 p^{20} T^{26} + 10752 p^{22} T^{27} - 676 p^{24} T^{28} - 336 p^{26} T^{29} - 34 p^{28} T^{30} + p^{32} T^{32} \)
7 \( 1 + 8 T + 32 T^{2} - 428 T^{3} - 2340 T^{4} - 6108 T^{5} + 117608 T^{6} - 188852 T^{7} + 261090 T^{8} - 1664636 p T^{9} + 5573688 p^{2} T^{10} - 2426336 p^{3} T^{11} - 3931244 p^{4} T^{12} - 6019560 p^{5} T^{13} + 4643080 p^{6} T^{14} + 4813420 p^{7} T^{15} + 9246675 p^{8} T^{16} + 4813420 p^{9} T^{17} + 4643080 p^{10} T^{18} - 6019560 p^{11} T^{19} - 3931244 p^{12} T^{20} - 2426336 p^{13} T^{21} + 5573688 p^{14} T^{22} - 1664636 p^{15} T^{23} + 261090 p^{16} T^{24} - 188852 p^{18} T^{25} + 117608 p^{20} T^{26} - 6108 p^{22} T^{27} - 2340 p^{24} T^{28} - 428 p^{26} T^{29} + 32 p^{28} T^{30} + 8 p^{30} T^{31} + p^{32} T^{32} \)
good11 \( ( 1 + 16 T - 322 T^{2} - 2984 T^{3} + 84470 T^{4} - 50260 T^{5} - 16142276 T^{6} + 91798084 T^{7} + 2040053636 T^{8} - 1281794744 p T^{9} - 13867581876 p T^{10} + 1166843885404 T^{11} + 10952808020900 T^{12} - 73850587314660 T^{13} - 2325064981249694 T^{14} + 2308470374815584 T^{15} + 380031413761461427 T^{16} + 2308470374815584 p^{2} T^{17} - 2325064981249694 p^{4} T^{18} - 73850587314660 p^{6} T^{19} + 10952808020900 p^{8} T^{20} + 1166843885404 p^{10} T^{21} - 13867581876 p^{13} T^{22} - 1281794744 p^{15} T^{23} + 2040053636 p^{16} T^{24} + 91798084 p^{18} T^{25} - 16142276 p^{20} T^{26} - 50260 p^{22} T^{27} + 84470 p^{24} T^{28} - 2984 p^{26} T^{29} - 322 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
13 \( ( 1 + 16 T + 128 T^{2} + 188 p T^{3} + 14316 T^{4} - 343380 T^{5} - 4339960 T^{6} - 110487700 T^{7} - 2516638446 T^{8} - 19922152156 T^{9} - 145570872648 T^{10} - 2758835887024 T^{11} - 409842977996 T^{12} + 496418351850600 T^{13} + 31710506382280 p^{2} T^{14} + 8059325024212700 p T^{15} + 2142629679076020195 T^{16} + 8059325024212700 p^{3} T^{17} + 31710506382280 p^{6} T^{18} + 496418351850600 p^{6} T^{19} - 409842977996 p^{8} T^{20} - 2758835887024 p^{10} T^{21} - 145570872648 p^{12} T^{22} - 19922152156 p^{14} T^{23} - 2516638446 p^{16} T^{24} - 110487700 p^{18} T^{25} - 4339960 p^{20} T^{26} - 343380 p^{22} T^{27} + 14316 p^{24} T^{28} + 188 p^{27} T^{29} + 128 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
17 \( 1 + 56 T + 1568 T^{2} + 21560 T^{3} - 40112 T^{4} - 8811572 T^{5} - 198135616 T^{6} - 1982747424 T^{7} + 11700324696 T^{8} + 993772764656 T^{9} + 25431415962568 T^{10} + 421993637314564 T^{11} + 3651160992412552 T^{12} - 39831914850219044 T^{13} - 2310125442796729152 T^{14} - 49580163585504631296 T^{15} - \)\(56\!\cdots\!70\)\( T^{16} + \)\(98\!\cdots\!64\)\( T^{17} + \)\(21\!\cdots\!64\)\( T^{18} + \)\(52\!\cdots\!12\)\( T^{19} + \)\(71\!\cdots\!56\)\( T^{20} + \)\(27\!\cdots\!64\)\( T^{21} - \)\(15\!\cdots\!20\)\( T^{22} - \)\(46\!\cdots\!36\)\( T^{23} - \)\(75\!\cdots\!52\)\( T^{24} - \)\(59\!\cdots\!88\)\( T^{25} + \)\(59\!\cdots\!36\)\( T^{26} + \)\(30\!\cdots\!24\)\( T^{27} + \)\(59\!\cdots\!00\)\( T^{28} + \)\(60\!\cdots\!04\)\( T^{29} - \)\(27\!\cdots\!48\)\( T^{30} - \)\(26\!\cdots\!84\)\( T^{31} - \)\(60\!\cdots\!01\)\( T^{32} - \)\(26\!\cdots\!84\)\( p^{2} T^{33} - \)\(27\!\cdots\!48\)\( p^{4} T^{34} + \)\(60\!\cdots\!04\)\( p^{6} T^{35} + \)\(59\!\cdots\!00\)\( p^{8} T^{36} + \)\(30\!\cdots\!24\)\( p^{10} T^{37} + \)\(59\!\cdots\!36\)\( p^{12} T^{38} - \)\(59\!\cdots\!88\)\( p^{14} T^{39} - \)\(75\!\cdots\!52\)\( p^{16} T^{40} - \)\(46\!\cdots\!36\)\( p^{18} T^{41} - \)\(15\!\cdots\!20\)\( p^{20} T^{42} + \)\(27\!\cdots\!64\)\( p^{22} T^{43} + \)\(71\!\cdots\!56\)\( p^{24} T^{44} + \)\(52\!\cdots\!12\)\( p^{26} T^{45} + \)\(21\!\cdots\!64\)\( p^{28} T^{46} + \)\(98\!\cdots\!64\)\( p^{30} T^{47} - \)\(56\!\cdots\!70\)\( p^{32} T^{48} - 49580163585504631296 p^{34} T^{49} - 2310125442796729152 p^{36} T^{50} - 39831914850219044 p^{38} T^{51} + 3651160992412552 p^{40} T^{52} + 421993637314564 p^{42} T^{53} + 25431415962568 p^{44} T^{54} + 993772764656 p^{46} T^{55} + 11700324696 p^{48} T^{56} - 1982747424 p^{50} T^{57} - 198135616 p^{52} T^{58} - 8811572 p^{54} T^{59} - 40112 p^{56} T^{60} + 21560 p^{58} T^{61} + 1568 p^{60} T^{62} + 56 p^{62} T^{63} + p^{64} T^{64} \)
19 \( 1 + 2344 T^{2} + 3033068 T^{4} + 2502523792 T^{6} + 1346852178486 T^{8} + 339036813313568 T^{10} - 143875083497505416 T^{12} - \)\(21\!\cdots\!56\)\( T^{14} - \)\(12\!\cdots\!23\)\( T^{16} - \)\(35\!\cdots\!40\)\( T^{18} + \)\(26\!\cdots\!76\)\( T^{20} + \)\(88\!\cdots\!52\)\( T^{22} + \)\(49\!\cdots\!30\)\( T^{24} + \)\(13\!\cdots\!84\)\( T^{26} - \)\(16\!\cdots\!76\)\( T^{28} - \)\(59\!\cdots\!48\)\( p^{2} T^{30} - \)\(84\!\cdots\!80\)\( p^{4} T^{32} - \)\(59\!\cdots\!48\)\( p^{6} T^{34} - \)\(16\!\cdots\!76\)\( p^{8} T^{36} + \)\(13\!\cdots\!84\)\( p^{12} T^{38} + \)\(49\!\cdots\!30\)\( p^{16} T^{40} + \)\(88\!\cdots\!52\)\( p^{20} T^{42} + \)\(26\!\cdots\!76\)\( p^{24} T^{44} - \)\(35\!\cdots\!40\)\( p^{28} T^{46} - \)\(12\!\cdots\!23\)\( p^{32} T^{48} - \)\(21\!\cdots\!56\)\( p^{36} T^{50} - 143875083497505416 p^{40} T^{52} + 339036813313568 p^{44} T^{54} + 1346852178486 p^{48} T^{56} + 2502523792 p^{52} T^{58} + 3033068 p^{56} T^{60} + 2344 p^{60} T^{62} + p^{64} T^{64} \)
23 \( 1 + 48 T + 1152 T^{2} - 7592 T^{3} - 672304 T^{4} - 13448884 T^{5} + 157767008 T^{6} + 5516634208 T^{7} - 53384392952 T^{8} - 6794784279760 T^{9} - 110029480685752 T^{10} + 2462050691124260 T^{11} + 166901127855959144 T^{12} + 173435548151687172 p T^{13} + 937338481100674720 p T^{14} - \)\(14\!\cdots\!48\)\( T^{15} - \)\(32\!\cdots\!26\)\( T^{16} + \)\(11\!\cdots\!68\)\( T^{17} + \)\(18\!\cdots\!92\)\( T^{18} + \)\(56\!\cdots\!24\)\( T^{19} - \)\(13\!\cdots\!28\)\( T^{20} - \)\(42\!\cdots\!68\)\( T^{21} - \)\(13\!\cdots\!88\)\( T^{22} + \)\(21\!\cdots\!64\)\( T^{23} + \)\(60\!\cdots\!52\)\( T^{24} + \)\(54\!\cdots\!32\)\( T^{25} - \)\(12\!\cdots\!80\)\( T^{26} - \)\(32\!\cdots\!40\)\( T^{27} + \)\(42\!\cdots\!72\)\( T^{28} + \)\(32\!\cdots\!28\)\( T^{29} + \)\(41\!\cdots\!28\)\( T^{30} - \)\(12\!\cdots\!16\)\( T^{31} - \)\(53\!\cdots\!37\)\( T^{32} - \)\(12\!\cdots\!16\)\( p^{2} T^{33} + \)\(41\!\cdots\!28\)\( p^{4} T^{34} + \)\(32\!\cdots\!28\)\( p^{6} T^{35} + \)\(42\!\cdots\!72\)\( p^{8} T^{36} - \)\(32\!\cdots\!40\)\( p^{10} T^{37} - \)\(12\!\cdots\!80\)\( p^{12} T^{38} + \)\(54\!\cdots\!32\)\( p^{14} T^{39} + \)\(60\!\cdots\!52\)\( p^{16} T^{40} + \)\(21\!\cdots\!64\)\( p^{18} T^{41} - \)\(13\!\cdots\!88\)\( p^{20} T^{42} - \)\(42\!\cdots\!68\)\( p^{22} T^{43} - \)\(13\!\cdots\!28\)\( p^{24} T^{44} + \)\(56\!\cdots\!24\)\( p^{26} T^{45} + \)\(18\!\cdots\!92\)\( p^{28} T^{46} + \)\(11\!\cdots\!68\)\( p^{30} T^{47} - \)\(32\!\cdots\!26\)\( p^{32} T^{48} - \)\(14\!\cdots\!48\)\( p^{34} T^{49} + 937338481100674720 p^{37} T^{50} + 173435548151687172 p^{39} T^{51} + 166901127855959144 p^{40} T^{52} + 2462050691124260 p^{42} T^{53} - 110029480685752 p^{44} T^{54} - 6794784279760 p^{46} T^{55} - 53384392952 p^{48} T^{56} + 5516634208 p^{50} T^{57} + 157767008 p^{52} T^{58} - 13448884 p^{54} T^{59} - 672304 p^{56} T^{60} - 7592 p^{58} T^{61} + 1152 p^{60} T^{62} + 48 p^{62} T^{63} + p^{64} T^{64} \)
29 \( ( 1 - 6720 T^{2} + 24117516 T^{4} - 60007636444 T^{6} + 114277499004756 T^{8} - 175249552287471988 T^{10} + \)\(22\!\cdots\!16\)\( T^{12} - \)\(23\!\cdots\!68\)\( T^{14} + \)\(21\!\cdots\!22\)\( T^{16} - \)\(23\!\cdots\!68\)\( p^{4} T^{18} + \)\(22\!\cdots\!16\)\( p^{8} T^{20} - 175249552287471988 p^{12} T^{22} + 114277499004756 p^{16} T^{24} - 60007636444 p^{20} T^{26} + 24117516 p^{24} T^{28} - 6720 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
31 \( ( 1 - 80 T + 1148 T^{2} + 48304 T^{3} - 1725690 T^{4} + 68427656 T^{5} - 1608024296 T^{6} - 65723247928 T^{7} + 81055630271 p T^{8} - 201838430704 T^{9} - 372303478086952 T^{10} + 7851188509333696 T^{11} + 809862074358801078 T^{12} - 70983157750195402520 T^{13} + \)\(38\!\cdots\!84\)\( T^{14} + \)\(63\!\cdots\!92\)\( T^{15} - \)\(24\!\cdots\!20\)\( T^{16} + \)\(63\!\cdots\!92\)\( p^{2} T^{17} + \)\(38\!\cdots\!84\)\( p^{4} T^{18} - 70983157750195402520 p^{6} T^{19} + 809862074358801078 p^{8} T^{20} + 7851188509333696 p^{10} T^{21} - 372303478086952 p^{12} T^{22} - 201838430704 p^{14} T^{23} + 81055630271 p^{17} T^{24} - 65723247928 p^{18} T^{25} - 1608024296 p^{20} T^{26} + 68427656 p^{22} T^{27} - 1725690 p^{24} T^{28} + 48304 p^{26} T^{29} + 1148 p^{28} T^{30} - 80 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
37 \( 1 - 44 T + 968 T^{2} + 80680 T^{3} - 6261832 T^{4} + 219964744 T^{5} - 362364160 T^{6} - 343125693056 T^{7} + 19932719523342 T^{8} - 15617733316640 p T^{9} + 370597233888432 T^{10} + 856798318641781236 T^{11} - 47401629410218598568 T^{12} + \)\(13\!\cdots\!08\)\( T^{13} - \)\(33\!\cdots\!56\)\( T^{14} - \)\(17\!\cdots\!20\)\( T^{15} + \)\(92\!\cdots\!29\)\( T^{16} - \)\(23\!\cdots\!28\)\( T^{17} - \)\(37\!\cdots\!20\)\( T^{18} + \)\(33\!\cdots\!72\)\( T^{19} - \)\(15\!\cdots\!48\)\( T^{20} + \)\(47\!\cdots\!76\)\( T^{21} - \)\(12\!\cdots\!96\)\( T^{22} - \)\(62\!\cdots\!12\)\( T^{23} + \)\(41\!\cdots\!74\)\( T^{24} - \)\(12\!\cdots\!52\)\( T^{25} + \)\(30\!\cdots\!12\)\( T^{26} + \)\(18\!\cdots\!04\)\( T^{27} - \)\(10\!\cdots\!76\)\( T^{28} + \)\(26\!\cdots\!96\)\( T^{29} + \)\(23\!\cdots\!92\)\( T^{30} - \)\(39\!\cdots\!08\)\( T^{31} + \)\(20\!\cdots\!00\)\( T^{32} - \)\(39\!\cdots\!08\)\( p^{2} T^{33} + \)\(23\!\cdots\!92\)\( p^{4} T^{34} + \)\(26\!\cdots\!96\)\( p^{6} T^{35} - \)\(10\!\cdots\!76\)\( p^{8} T^{36} + \)\(18\!\cdots\!04\)\( p^{10} T^{37} + \)\(30\!\cdots\!12\)\( p^{12} T^{38} - \)\(12\!\cdots\!52\)\( p^{14} T^{39} + \)\(41\!\cdots\!74\)\( p^{16} T^{40} - \)\(62\!\cdots\!12\)\( p^{18} T^{41} - \)\(12\!\cdots\!96\)\( p^{20} T^{42} + \)\(47\!\cdots\!76\)\( p^{22} T^{43} - \)\(15\!\cdots\!48\)\( p^{24} T^{44} + \)\(33\!\cdots\!72\)\( p^{26} T^{45} - \)\(37\!\cdots\!20\)\( p^{28} T^{46} - \)\(23\!\cdots\!28\)\( p^{30} T^{47} + \)\(92\!\cdots\!29\)\( p^{32} T^{48} - \)\(17\!\cdots\!20\)\( p^{34} T^{49} - \)\(33\!\cdots\!56\)\( p^{36} T^{50} + \)\(13\!\cdots\!08\)\( p^{38} T^{51} - 47401629410218598568 p^{40} T^{52} + 856798318641781236 p^{42} T^{53} + 370597233888432 p^{44} T^{54} - 15617733316640 p^{47} T^{55} + 19932719523342 p^{48} T^{56} - 343125693056 p^{50} T^{57} - 362364160 p^{52} T^{58} + 219964744 p^{54} T^{59} - 6261832 p^{56} T^{60} + 80680 p^{58} T^{61} + 968 p^{60} T^{62} - 44 p^{62} T^{63} + p^{64} T^{64} \)
41 \( ( 1 + 20 T + 1278 T^{2} - 38276 T^{3} + 853934 T^{4} + 4541436 T^{5} + 5875241188 T^{6} - 27274394992 T^{7} + 4538409861312 T^{8} - 27274394992 p^{2} T^{9} + 5875241188 p^{4} T^{10} + 4541436 p^{6} T^{11} + 853934 p^{8} T^{12} - 38276 p^{10} T^{13} + 1278 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} )^{4} \)
43 \( ( 1 + 92 T + 4232 T^{2} + 234308 T^{3} + 17156192 T^{4} + 850806472 T^{5} + 33119310312 T^{6} + 1548360435128 T^{7} + 57744913080946 T^{8} + 1516830516564260 T^{9} + 51699751977611360 T^{10} + 1998662968759146740 T^{11} + 4648074954869677772 T^{12} - \)\(25\!\cdots\!36\)\( T^{13} - \)\(99\!\cdots\!56\)\( T^{14} - \)\(44\!\cdots\!64\)\( T^{15} - \)\(19\!\cdots\!41\)\( T^{16} - \)\(44\!\cdots\!64\)\( p^{2} T^{17} - \)\(99\!\cdots\!56\)\( p^{4} T^{18} - \)\(25\!\cdots\!36\)\( p^{6} T^{19} + 4648074954869677772 p^{8} T^{20} + 1998662968759146740 p^{10} T^{21} + 51699751977611360 p^{12} T^{22} + 1516830516564260 p^{14} T^{23} + 57744913080946 p^{16} T^{24} + 1548360435128 p^{18} T^{25} + 33119310312 p^{20} T^{26} + 850806472 p^{22} T^{27} + 17156192 p^{24} T^{28} + 234308 p^{26} T^{29} + 4232 p^{28} T^{30} + 92 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
47 \( 1 + 228 T + 25992 T^{2} + 1929672 T^{3} + 102499780 T^{4} + 3910376532 T^{5} + 89208581328 T^{6} - 713393385972 T^{7} - 179847461096628 T^{8} - 9445294864601736 T^{9} - 353884911424936344 T^{10} - 14505371407282045524 T^{11} - \)\(46\!\cdots\!44\)\( T^{12} + \)\(25\!\cdots\!84\)\( T^{13} + \)\(54\!\cdots\!16\)\( T^{14} + \)\(45\!\cdots\!96\)\( T^{15} + \)\(24\!\cdots\!38\)\( T^{16} + \)\(88\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!00\)\( T^{18} - \)\(10\!\cdots\!40\)\( T^{19} - \)\(83\!\cdots\!72\)\( T^{20} - \)\(31\!\cdots\!16\)\( T^{21} - \)\(41\!\cdots\!04\)\( T^{22} + \)\(24\!\cdots\!16\)\( T^{23} + \)\(23\!\cdots\!12\)\( T^{24} + \)\(15\!\cdots\!72\)\( T^{25} + \)\(10\!\cdots\!68\)\( T^{26} + \)\(64\!\cdots\!68\)\( T^{27} + \)\(31\!\cdots\!60\)\( T^{28} + \)\(10\!\cdots\!08\)\( T^{29} + \)\(62\!\cdots\!32\)\( T^{30} - \)\(18\!\cdots\!68\)\( T^{31} - \)\(13\!\cdots\!93\)\( T^{32} - \)\(18\!\cdots\!68\)\( p^{2} T^{33} + \)\(62\!\cdots\!32\)\( p^{4} T^{34} + \)\(10\!\cdots\!08\)\( p^{6} T^{35} + \)\(31\!\cdots\!60\)\( p^{8} T^{36} + \)\(64\!\cdots\!68\)\( p^{10} T^{37} + \)\(10\!\cdots\!68\)\( p^{12} T^{38} + \)\(15\!\cdots\!72\)\( p^{14} T^{39} + \)\(23\!\cdots\!12\)\( p^{16} T^{40} + \)\(24\!\cdots\!16\)\( p^{18} T^{41} - \)\(41\!\cdots\!04\)\( p^{20} T^{42} - \)\(31\!\cdots\!16\)\( p^{22} T^{43} - \)\(83\!\cdots\!72\)\( p^{24} T^{44} - \)\(10\!\cdots\!40\)\( p^{26} T^{45} + \)\(10\!\cdots\!00\)\( p^{28} T^{46} + \)\(88\!\cdots\!00\)\( p^{30} T^{47} + \)\(24\!\cdots\!38\)\( p^{32} T^{48} + \)\(45\!\cdots\!96\)\( p^{34} T^{49} + \)\(54\!\cdots\!16\)\( p^{36} T^{50} + \)\(25\!\cdots\!84\)\( p^{38} T^{51} - \)\(46\!\cdots\!44\)\( p^{40} T^{52} - 14505371407282045524 p^{42} T^{53} - 353884911424936344 p^{44} T^{54} - 9445294864601736 p^{46} T^{55} - 179847461096628 p^{48} T^{56} - 713393385972 p^{50} T^{57} + 89208581328 p^{52} T^{58} + 3910376532 p^{54} T^{59} + 102499780 p^{56} T^{60} + 1929672 p^{58} T^{61} + 25992 p^{60} T^{62} + 228 p^{62} T^{63} + p^{64} T^{64} \)
53 \( 1 - 48 T + 1152 T^{2} - 26464 T^{3} + 12159320 T^{4} - 392146736 T^{5} + 5165678336 T^{6} + 489888623792 T^{7} + 81656780093668 T^{8} - 3756641219420384 T^{9} + 87364150697476736 T^{10} + 1265392886696245552 T^{11} + \)\(96\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!56\)\( T^{13} + \)\(17\!\cdots\!40\)\( T^{14} - \)\(11\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} - \)\(33\!\cdots\!00\)\( T^{17} + \)\(70\!\cdots\!64\)\( T^{18} - \)\(61\!\cdots\!20\)\( T^{19} + \)\(13\!\cdots\!12\)\( T^{20} - \)\(35\!\cdots\!64\)\( T^{21} + \)\(19\!\cdots\!88\)\( p T^{22} - \)\(29\!\cdots\!68\)\( T^{23} + \)\(97\!\cdots\!76\)\( T^{24} - \)\(22\!\cdots\!96\)\( T^{25} + \)\(16\!\cdots\!40\)\( T^{26} - \)\(38\!\cdots\!68\)\( T^{27} + \)\(85\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!84\)\( T^{29} + \)\(60\!\cdots\!76\)\( T^{30} - \)\(30\!\cdots\!92\)\( T^{31} + \)\(65\!\cdots\!63\)\( T^{32} - \)\(30\!\cdots\!92\)\( p^{2} T^{33} + \)\(60\!\cdots\!76\)\( p^{4} T^{34} - \)\(19\!\cdots\!84\)\( p^{6} T^{35} + \)\(85\!\cdots\!64\)\( p^{8} T^{36} - \)\(38\!\cdots\!68\)\( p^{10} T^{37} + \)\(16\!\cdots\!40\)\( p^{12} T^{38} - \)\(22\!\cdots\!96\)\( p^{14} T^{39} + \)\(97\!\cdots\!76\)\( p^{16} T^{40} - \)\(29\!\cdots\!68\)\( p^{18} T^{41} + \)\(19\!\cdots\!88\)\( p^{21} T^{42} - \)\(35\!\cdots\!64\)\( p^{22} T^{43} + \)\(13\!\cdots\!12\)\( p^{24} T^{44} - \)\(61\!\cdots\!20\)\( p^{26} T^{45} + \)\(70\!\cdots\!64\)\( p^{28} T^{46} - \)\(33\!\cdots\!00\)\( p^{30} T^{47} + \)\(11\!\cdots\!26\)\( p^{32} T^{48} - \)\(11\!\cdots\!20\)\( p^{34} T^{49} + \)\(17\!\cdots\!40\)\( p^{36} T^{50} - \)\(25\!\cdots\!56\)\( p^{38} T^{51} + \)\(96\!\cdots\!04\)\( p^{40} T^{52} + 1265392886696245552 p^{42} T^{53} + 87364150697476736 p^{44} T^{54} - 3756641219420384 p^{46} T^{55} + 81656780093668 p^{48} T^{56} + 489888623792 p^{50} T^{57} + 5165678336 p^{52} T^{58} - 392146736 p^{54} T^{59} + 12159320 p^{56} T^{60} - 26464 p^{58} T^{61} + 1152 p^{60} T^{62} - 48 p^{62} T^{63} + p^{64} T^{64} \)
59 \( 1 + 32408 T^{2} + 549062548 T^{4} + 6456694139464 T^{6} + 58949840848204124 T^{8} + \)\(44\!\cdots\!92\)\( T^{10} + \)\(28\!\cdots\!60\)\( T^{12} + \)\(16\!\cdots\!08\)\( T^{14} + \)\(79\!\cdots\!46\)\( T^{16} + \)\(35\!\cdots\!36\)\( T^{18} + \)\(14\!\cdots\!88\)\( T^{20} + \)\(51\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!72\)\( T^{24} + \)\(50\!\cdots\!64\)\( T^{26} + \)\(14\!\cdots\!24\)\( T^{28} + \)\(38\!\cdots\!84\)\( T^{30} + \)\(12\!\cdots\!91\)\( T^{32} + \)\(38\!\cdots\!84\)\( p^{4} T^{34} + \)\(14\!\cdots\!24\)\( p^{8} T^{36} + \)\(50\!\cdots\!64\)\( p^{12} T^{38} + \)\(16\!\cdots\!72\)\( p^{16} T^{40} + \)\(51\!\cdots\!60\)\( p^{20} T^{42} + \)\(14\!\cdots\!88\)\( p^{24} T^{44} + \)\(35\!\cdots\!36\)\( p^{28} T^{46} + \)\(79\!\cdots\!46\)\( p^{32} T^{48} + \)\(16\!\cdots\!08\)\( p^{36} T^{50} + \)\(28\!\cdots\!60\)\( p^{40} T^{52} + \)\(44\!\cdots\!92\)\( p^{44} T^{54} + 58949840848204124 p^{48} T^{56} + 6456694139464 p^{52} T^{58} + 549062548 p^{56} T^{60} + 32408 p^{60} T^{62} + p^{64} T^{64} \)
61 \( ( 1 - 108 T + 3344 T^{2} - 261456 T^{3} + 7251308 T^{4} + 1036857120 T^{5} + 18687170776 T^{6} - 210406697556 T^{7} - 304817639593378 T^{8} + 6951122192324784 T^{9} - 1315812464339416844 T^{10} + 76473784330953800400 T^{11} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(85\!\cdots\!12\)\( T^{13} + \)\(45\!\cdots\!52\)\( T^{14} - \)\(38\!\cdots\!20\)\( T^{15} - \)\(43\!\cdots\!33\)\( T^{16} - \)\(38\!\cdots\!20\)\( p^{2} T^{17} + \)\(45\!\cdots\!52\)\( p^{4} T^{18} - \)\(85\!\cdots\!12\)\( p^{6} T^{19} + \)\(22\!\cdots\!00\)\( p^{8} T^{20} + 76473784330953800400 p^{10} T^{21} - 1315812464339416844 p^{12} T^{22} + 6951122192324784 p^{14} T^{23} - 304817639593378 p^{16} T^{24} - 210406697556 p^{18} T^{25} + 18687170776 p^{20} T^{26} + 1036857120 p^{22} T^{27} + 7251308 p^{24} T^{28} - 261456 p^{26} T^{29} + 3344 p^{28} T^{30} - 108 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
67 \( 1 + 3.85e5T^{3} + 3.38e7T^{4} + 2.72e9T^{5} + 7.41e10T^{6} + 2.09e13T^{7} + 1.53e15T^{8} + 8.23e16T^{9} + 9.24e18T^{10} + 6.75e20T^{11} + 6.56e22T^{12} + 3.13e24T^{13} + 2.76e26T^{14} + 2.47e28T^{15} + 1.70e30T^{16} + 1.24e32T^{17} + 7.20e33T^{18} + 6.58e35T^{19} + 4.22e37T^{20} + 2.93e39T^{21} + 1.96e41T^{22} + 1.40e43T^{23} + 1.10e45T^{24} + 5.83e46T^{25} + 4.40e48T^{26} + 2.81e50T^{27} + 2.28e52T^{28} + 1.40e54T^{29} + 7.97e55T^{30} + 6.60e57T^{31}+O(T^{32}) \)
71 \( 1 - 368T + 1.46e5T^{2} - 3.84e7T^{3} + 9.57e9T^{4} - 2.00e12T^{5} + 3.90e14T^{6} - 6.90e16T^{7} + 1.14e19T^{8} - 1.77e21T^{9} + 2.60e23T^{10} - 3.62e25T^{11} + 4.80e27T^{12} - 6.09e29T^{13} + 7.41e31T^{14} - 8.68e33T^{15} + 9.79e35T^{16} - 1.06e38T^{17} + 1.12e40T^{18} - 1.14e42T^{19} + 1.13e44T^{20} - 1.09e46T^{21} + 1.01e48T^{22} - 9.23e49T^{23} + 8.15e51T^{24} - 7.00e53T^{25} + 5.86e55T^{26} - 4.78e57T^{27} + 3.80e59T^{28} - 2.94e61T^{29} + 2.22e63T^{30} - 1.64e65T^{31}+O(T^{32}) \)
73 \( 1 - 52T + 1.35e3T^{2} + 1.65e6T^{3} - 8.01e7T^{4} + 4.24e9T^{5} + 1.26e12T^{6} - 6.27e13T^{7} + 5.24e15T^{8} + 5.66e17T^{9} - 3.04e19T^{10} + 3.88e21T^{11} + 1.63e23T^{12} - 1.18e25T^{13} + 2.13e27T^{14} + 4.64e28T^{15} - 5.57e30T^{16} + 1.02e33T^{17} + 2.21e34T^{18} - 2.88e36T^{19} + 4.91e38T^{20} + 9.52e39T^{21} - 1.12e42T^{22} + 2.28e44T^{23} + 1.78e45T^{24} - 2.42e47T^{25} + 9.83e49T^{26} - 4.35e50T^{27} - 1.90e52T^{28} + 3.90e55T^{29} - 2.75e56T^{30}+O(T^{31}) \)
79 \( 1 + 7.31e4T^{2} + 2.74e9T^{4} + 7.02e13T^{6} + 1.38e18T^{8} + 2.25e22T^{10} + 3.12e26T^{12} + 3.81e30T^{14} + 4.19e34T^{16} + 4.20e38T^{18} + 3.88e42T^{20} + 3.34e46T^{22} + 2.68e50T^{24} + 2.03e54T^{26} + 1.45e58T^{28} + 9.85e61T^{30}+O(T^{31}) \)
83 \( 1 + 736T + 2.70e5T^{2} + 6.83e7T^{3} + 1.37e10T^{4} + 2.41e12T^{5} + 3.80e14T^{6} + 5.48e16T^{7} + 7.29e18T^{8} + 9.10e20T^{9} + 1.07e23T^{10} + 1.20e25T^{11} + 1.30e27T^{12} + 1.36e29T^{13} + 1.37e31T^{14} + 1.36e33T^{15} + 1.33e35T^{16} + 1.27e37T^{17} + 1.21e39T^{18} + 1.14e41T^{19} + 1.05e43T^{20} + 9.74e44T^{21} + 8.85e46T^{22} + 7.97e48T^{23} + 7.11e50T^{24} + 6.29e52T^{25} + 5.52e54T^{26} + 4.82e56T^{27} + 4.17e58T^{28} + 3.58e60T^{29} + 3.05e62T^{30}+O(T^{31}) \)
89 \( 1 + 1.07e5T^{2} + 6.06e9T^{4} + 2.34e14T^{6} + 7.03e18T^{8} + 1.72e23T^{10} + 3.61e27T^{12} + 6.58e31T^{14} + 1.06e36T^{16} + 1.55e40T^{18} + 2.05e44T^{20} + 2.48e48T^{22} + 2.76e52T^{24} + 2.84e56T^{26} + 2.70e60T^{28}+O(T^{30}) \)
97 \( 1 + 408T + 8.32e4T^{2} + 1.62e7T^{3} + 3.12e9T^{4} + 4.86e11T^{5} + 7.02e13T^{6} + 1.03e16T^{7} + 1.39e18T^{8} + 1.80e20T^{9} + 2.36e22T^{10} + 3.00e24T^{11} + 3.69e26T^{12} + 4.53e28T^{13} + 5.46e30T^{14} + 6.43e32T^{15} + 7.47e34T^{16} + 8.55e36T^{17} + 9.67e38T^{18} + 1.07e41T^{19} + 1.18e43T^{20} + 1.29e45T^{21} + 1.40e47T^{22} + 1.49e49T^{23} + 1.57e51T^{24} + 1.64e53T^{25} + 1.69e55T^{26} + 1.73e57T^{27} + 1.75e59T^{28} + 1.76e61T^{29}+O(T^{30}) \)
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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

−1.95205427920113062310385271721, −1.94649717581973929963504941344, −1.92188620600466867750849031218, −1.80601220009932819551524143855, −1.70345254461784551357601521262, −1.66991945306836982563838329990, −1.56102405399834887693312593588, −1.39077222409628447125624509915, −1.33891294561774971533957829076, −1.26771308127134281766494682040, −1.22713533517475452257480073993, −1.19384467175426224165255249324, −1.19357646452423207889704678801, −0.979287620844757080949829086384, −0.69645499207648695387682237822, −0.64762672785903059917842802202, −0.64379541355079312629408121681, −0.56555413309403281794656046532, −0.48328304602846934425516382099, −0.46246401457790067305604490177, −0.43194879121368124789647580515, −0.34628223369835469488790338967, −0.29764482132771626627806136866, −0.28078621828448502309068435699, −0.26357765092275244725524742312, 0.26357765092275244725524742312, 0.28078621828448502309068435699, 0.29764482132771626627806136866, 0.34628223369835469488790338967, 0.43194879121368124789647580515, 0.46246401457790067305604490177, 0.48328304602846934425516382099, 0.56555413309403281794656046532, 0.64379541355079312629408121681, 0.64762672785903059917842802202, 0.69645499207648695387682237822, 0.979287620844757080949829086384, 1.19357646452423207889704678801, 1.19384467175426224165255249324, 1.22713533517475452257480073993, 1.26771308127134281766494682040, 1.33891294561774971533957829076, 1.39077222409628447125624509915, 1.56102405399834887693312593588, 1.66991945306836982563838329990, 1.70345254461784551357601521262, 1.80601220009932819551524143855, 1.92188620600466867750849031218, 1.94649717581973929963504941344, 1.95205427920113062310385271721

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

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