Properties

Degree 96
Conductor $ 3^{48} \cdot 5^{48} \cdot 7^{48} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 12·7-s + 2·9-s − 16·13-s − 10·16-s + 24·21-s + 2·25-s + 12·27-s − 24·31-s + 4·37-s + 32·39-s + 16·43-s + 20·48-s + 72·49-s − 8·61-s − 24·63-s + 12·67-s + 52·73-s − 4·75-s − 21·81-s + 192·91-s + 48·93-s − 120·97-s + 40·103-s − 8·111-s + 120·112-s − 32·117-s + ⋯
L(s)  = 1  − 1.15·3-s − 4.53·7-s + 2/3·9-s − 4.43·13-s − 5/2·16-s + 5.23·21-s + 2/5·25-s + 2.30·27-s − 4.31·31-s + 0.657·37-s + 5.12·39-s + 2.43·43-s + 2.88·48-s + 72/7·49-s − 1.02·61-s − 3.02·63-s + 1.46·67-s + 6.08·73-s − 0.461·75-s − 7/3·81-s + 20.1·91-s + 4.97·93-s − 12.1·97-s + 3.94·103-s − 0.759·111-s + 11.3·112-s − 2.95·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(96\)
\( N \)  =  \(3^{48} \cdot 5^{48} \cdot 7^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(96,\ 3^{48} \cdot 5^{48} \cdot 7^{48} ,\ ( \ : [1/2]^{48} ),\ 1 )$
$L(1)$  $\approx$  $0.0397887$
$L(\frac12)$  $\approx$  $0.0397887$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 96. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 95.
$p$$F_p(T)$
bad3 \( 1 + 2 T + 2 T^{2} - 4 p T^{3} - 31 T^{4} - 28 T^{5} + 26 p T^{6} + 178 T^{7} + 122 T^{8} - 646 T^{9} - 862 T^{10} + 260 T^{11} + 1709 p T^{12} + 652 p^{2} T^{13} - 3946 T^{14} - 33814 T^{15} - 30344 T^{16} + 2914 p^{2} T^{17} + 21386 p^{2} T^{18} + 17084 p^{2} T^{19} - 179267 T^{20} - 963508 T^{21} - 689290 T^{22} + 914882 T^{23} + 4875904 T^{24} + 914882 p T^{25} - 689290 p^{2} T^{26} - 963508 p^{3} T^{27} - 179267 p^{4} T^{28} + 17084 p^{7} T^{29} + 21386 p^{8} T^{30} + 2914 p^{9} T^{31} - 30344 p^{8} T^{32} - 33814 p^{9} T^{33} - 3946 p^{10} T^{34} + 652 p^{13} T^{35} + 1709 p^{13} T^{36} + 260 p^{13} T^{37} - 862 p^{14} T^{38} - 646 p^{15} T^{39} + 122 p^{16} T^{40} + 178 p^{17} T^{41} + 26 p^{19} T^{42} - 28 p^{19} T^{43} - 31 p^{20} T^{44} - 4 p^{22} T^{45} + 2 p^{22} T^{46} + 2 p^{23} T^{47} + p^{24} T^{48} \)
5 \( 1 - 2 T^{2} + 14 T^{4} + 72 T^{6} - 1351 T^{8} + 4396 T^{10} - 16534 T^{12} - 73794 T^{14} + 1049466 T^{16} - 669814 p T^{18} + 1314142 p T^{20} + 2757756 p^{2} T^{22} - 25812791 p^{2} T^{24} + 2757756 p^{4} T^{26} + 1314142 p^{5} T^{28} - 669814 p^{7} T^{30} + 1049466 p^{8} T^{32} - 73794 p^{10} T^{34} - 16534 p^{12} T^{36} + 4396 p^{14} T^{38} - 1351 p^{16} T^{40} + 72 p^{18} T^{42} + 14 p^{20} T^{44} - 2 p^{22} T^{46} + p^{24} T^{48} \)
7 \( ( 1 + 6 T + 18 T^{2} + 54 T^{3} + 201 T^{4} + 690 T^{5} + 1980 T^{6} + 5622 T^{7} + 18390 T^{8} + 57006 T^{9} + 154674 T^{10} + 424266 T^{11} + 1154801 T^{12} + 424266 p T^{13} + 154674 p^{2} T^{14} + 57006 p^{3} T^{15} + 18390 p^{4} T^{16} + 5622 p^{5} T^{17} + 1980 p^{6} T^{18} + 690 p^{7} T^{19} + 201 p^{8} T^{20} + 54 p^{9} T^{21} + 18 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
good2 \( 1 + 5 p T^{4} + 31 T^{8} + 93 p T^{12} + 1553 T^{16} + 63 p^{6} T^{20} + 377 p^{5} T^{24} + 3889 p^{5} T^{28} + 80551 p^{2} T^{32} + 12585 p^{5} T^{36} + 498205 p^{4} T^{40} + 1673229 p^{4} T^{44} + 819905 p^{4} T^{48} + 1673229 p^{8} T^{52} + 498205 p^{12} T^{56} + 12585 p^{17} T^{60} + 80551 p^{18} T^{64} + 3889 p^{25} T^{68} + 377 p^{29} T^{72} + 63 p^{34} T^{76} + 1553 p^{32} T^{80} + 93 p^{37} T^{84} + 31 p^{40} T^{88} + 5 p^{45} T^{92} + p^{48} T^{96} \)
11 \( ( 1 + 82 T^{2} + 28 p^{2} T^{4} + 93648 T^{6} + 1960811 T^{8} + 33901866 T^{10} + 522006868 T^{12} + 7579610630 T^{14} + 9610234148 p T^{16} + 1395615342912 T^{18} + 17189621822992 T^{20} + 199045695692190 T^{22} + 2217712891618049 T^{24} + 199045695692190 p^{2} T^{26} + 17189621822992 p^{4} T^{28} + 1395615342912 p^{6} T^{30} + 9610234148 p^{9} T^{32} + 7579610630 p^{10} T^{34} + 522006868 p^{12} T^{36} + 33901866 p^{14} T^{38} + 1960811 p^{16} T^{40} + 93648 p^{18} T^{42} + 28 p^{22} T^{44} + 82 p^{22} T^{46} + p^{24} T^{48} )^{2} \)
13 \( ( 1 + 4 T + 8 T^{2} + 6 p T^{3} + 412 T^{4} + 126 T^{5} + 250 T^{6} + 4046 T^{7} - 8472 T^{8} - 115998 T^{9} - 175690 T^{10} - 787608 T^{11} - 578230 T^{12} - 787608 p T^{13} - 175690 p^{2} T^{14} - 115998 p^{3} T^{15} - 8472 p^{4} T^{16} + 4046 p^{5} T^{17} + 250 p^{6} T^{18} + 126 p^{7} T^{19} + 412 p^{8} T^{20} + 6 p^{10} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} )^{4} \)
17 \( 1 - 292 T^{4} - 162094 T^{8} + 26525208 T^{12} + 20334482521 T^{16} - 918830295452 T^{20} - 952044342289378 T^{24} - 238206196887158972 T^{28} - 27138094435460317050 T^{32} + \)\(23\!\cdots\!60\)\( T^{36} + \)\(13\!\cdots\!14\)\( T^{40} - \)\(11\!\cdots\!32\)\( T^{44} - \)\(13\!\cdots\!99\)\( T^{48} - \)\(11\!\cdots\!32\)\( p^{4} T^{52} + \)\(13\!\cdots\!14\)\( p^{8} T^{56} + \)\(23\!\cdots\!60\)\( p^{12} T^{60} - 27138094435460317050 p^{16} T^{64} - 238206196887158972 p^{20} T^{68} - 952044342289378 p^{24} T^{72} - 918830295452 p^{28} T^{76} + 20334482521 p^{32} T^{80} + 26525208 p^{36} T^{84} - 162094 p^{40} T^{88} - 292 p^{44} T^{92} + p^{48} T^{96} \)
19 \( ( 1 + 156 T^{2} + 12884 T^{4} + 724240 T^{6} + 30596336 T^{8} + 1018616832 T^{10} + 27328536516 T^{12} + 592260316344 T^{14} + 10107438153448 T^{16} + 123059692662528 T^{18} + 597914468822404 T^{20} - 16379314282711204 T^{22} - 515694405066348730 T^{24} - 16379314282711204 p^{2} T^{26} + 597914468822404 p^{4} T^{28} + 123059692662528 p^{6} T^{30} + 10107438153448 p^{8} T^{32} + 592260316344 p^{10} T^{34} + 27328536516 p^{12} T^{36} + 1018616832 p^{14} T^{38} + 30596336 p^{16} T^{40} + 724240 p^{18} T^{42} + 12884 p^{20} T^{44} + 156 p^{22} T^{46} + p^{24} T^{48} )^{2} \)
23 \( 1 - 1258 T^{4} + 87961 T^{8} - 190569570 T^{12} + 689060279762 T^{16} - 168036410103954 T^{20} - 10104334963177457 T^{24} - \)\(17\!\cdots\!38\)\( T^{28} + \)\(69\!\cdots\!00\)\( T^{32} + \)\(58\!\cdots\!38\)\( T^{36} + \)\(31\!\cdots\!73\)\( T^{40} - \)\(15\!\cdots\!78\)\( T^{44} - \)\(10\!\cdots\!16\)\( T^{48} - \)\(15\!\cdots\!78\)\( p^{4} T^{52} + \)\(31\!\cdots\!73\)\( p^{8} T^{56} + \)\(58\!\cdots\!38\)\( p^{12} T^{60} + \)\(69\!\cdots\!00\)\( p^{16} T^{64} - \)\(17\!\cdots\!38\)\( p^{20} T^{68} - 10104334963177457 p^{24} T^{72} - 168036410103954 p^{28} T^{76} + 689060279762 p^{32} T^{80} - 190569570 p^{36} T^{84} + 87961 p^{40} T^{88} - 1258 p^{44} T^{92} + p^{48} T^{96} \)
29 \( ( 1 + 266 T^{2} + 34157 T^{4} + 2794800 T^{6} + 162053611 T^{8} + 7007937328 T^{10} + 231616105004 T^{12} + 7007937328 p^{2} T^{14} + 162053611 p^{4} T^{16} + 2794800 p^{6} T^{18} + 34157 p^{8} T^{20} + 266 p^{10} T^{22} + p^{12} T^{24} )^{4} \)
31 \( ( 1 + 6 T - 102 T^{2} - 660 T^{3} + 5810 T^{4} + 37790 T^{5} - 222836 T^{6} - 1334722 T^{7} + 220170 p T^{8} + 29859212 T^{9} - 200779810 T^{10} - 330993114 T^{11} + 5988664294 T^{12} - 330993114 p T^{13} - 200779810 p^{2} T^{14} + 29859212 p^{3} T^{15} + 220170 p^{5} T^{16} - 1334722 p^{5} T^{17} - 222836 p^{6} T^{18} + 37790 p^{7} T^{19} + 5810 p^{8} T^{20} - 660 p^{9} T^{21} - 102 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} )^{4} \)
37 \( ( 1 - 2 T + 2 T^{2} + 240 T^{3} + 340 T^{4} + 4412 T^{5} + 19296 T^{6} + 295596 T^{7} - 488500 T^{8} + 7135028 T^{9} + 60822216 T^{10} + 44332694 T^{11} + 5614623308 T^{12} - 17048319442 T^{13} + 2633027636 p T^{14} + 1520582252656 T^{15} - 347161403352 T^{16} + 29025433822024 T^{17} + 159173283221960 T^{18} + 1404254167923616 T^{19} - 2420759446723516 T^{20} + 64217666952951020 T^{21} + 328984624727805402 T^{22} - 724974596893577714 T^{23} + 17132717045972633682 T^{24} - 724974596893577714 p T^{25} + 328984624727805402 p^{2} T^{26} + 64217666952951020 p^{3} T^{27} - 2420759446723516 p^{4} T^{28} + 1404254167923616 p^{5} T^{29} + 159173283221960 p^{6} T^{30} + 29025433822024 p^{7} T^{31} - 347161403352 p^{8} T^{32} + 1520582252656 p^{9} T^{33} + 2633027636 p^{11} T^{34} - 17048319442 p^{11} T^{35} + 5614623308 p^{12} T^{36} + 44332694 p^{13} T^{37} + 60822216 p^{14} T^{38} + 7135028 p^{15} T^{39} - 488500 p^{16} T^{40} + 295596 p^{17} T^{41} + 19296 p^{18} T^{42} + 4412 p^{19} T^{43} + 340 p^{20} T^{44} + 240 p^{21} T^{45} + 2 p^{22} T^{46} - 2 p^{23} T^{47} + p^{24} T^{48} )^{2} \)
41 \( ( 1 - 316 T^{2} + 48097 T^{4} - 4708108 T^{6} + 334906659 T^{8} - 18580076262 T^{10} + 838484851868 T^{12} - 18580076262 p^{2} T^{14} + 334906659 p^{4} T^{16} - 4708108 p^{6} T^{18} + 48097 p^{8} T^{20} - 316 p^{10} T^{22} + p^{12} T^{24} )^{4} \)
43 \( ( 1 - 4 T + 8 T^{2} + 110 T^{3} - 519 T^{4} + 1910 T^{5} + 2562 T^{6} + 716292 T^{7} - 2637082 T^{8} - 20183148 T^{9} + 183775096 T^{10} - 402279870 T^{11} + 657997537 T^{12} - 402279870 p T^{13} + 183775096 p^{2} T^{14} - 20183148 p^{3} T^{15} - 2637082 p^{4} T^{16} + 716292 p^{5} T^{17} + 2562 p^{6} T^{18} + 1910 p^{7} T^{19} - 519 p^{8} T^{20} + 110 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} )^{4} \)
47 \( 1 + 2504 T^{4} - 9437418 T^{8} - 44486806016 T^{12} + 13141409366037 T^{16} + 333463418052279920 T^{20} + \)\(41\!\cdots\!26\)\( T^{24} - \)\(12\!\cdots\!64\)\( T^{28} - \)\(37\!\cdots\!86\)\( T^{32} + \)\(14\!\cdots\!08\)\( T^{36} + \)\(16\!\cdots\!62\)\( T^{40} + \)\(25\!\cdots\!48\)\( T^{44} - \)\(66\!\cdots\!75\)\( T^{48} + \)\(25\!\cdots\!48\)\( p^{4} T^{52} + \)\(16\!\cdots\!62\)\( p^{8} T^{56} + \)\(14\!\cdots\!08\)\( p^{12} T^{60} - \)\(37\!\cdots\!86\)\( p^{16} T^{64} - \)\(12\!\cdots\!64\)\( p^{20} T^{68} + \)\(41\!\cdots\!26\)\( p^{24} T^{72} + 333463418052279920 p^{28} T^{76} + 13141409366037 p^{32} T^{80} - 44486806016 p^{36} T^{84} - 9437418 p^{40} T^{88} + 2504 p^{44} T^{92} + p^{48} T^{96} \)
53 \( 1 + 3644 T^{4} + 5390702 T^{8} - 179789984 p T^{12} - 77048096040411 T^{16} - 108064890838368504 T^{20} + \)\(71\!\cdots\!90\)\( T^{24} + \)\(26\!\cdots\!88\)\( T^{28} + \)\(28\!\cdots\!42\)\( T^{32} - \)\(48\!\cdots\!24\)\( T^{36} - \)\(35\!\cdots\!90\)\( T^{40} - \)\(28\!\cdots\!64\)\( T^{44} + \)\(22\!\cdots\!77\)\( T^{48} - \)\(28\!\cdots\!64\)\( p^{4} T^{52} - \)\(35\!\cdots\!90\)\( p^{8} T^{56} - \)\(48\!\cdots\!24\)\( p^{12} T^{60} + \)\(28\!\cdots\!42\)\( p^{16} T^{64} + \)\(26\!\cdots\!88\)\( p^{20} T^{68} + \)\(71\!\cdots\!90\)\( p^{24} T^{72} - 108064890838368504 p^{28} T^{76} - 77048096040411 p^{32} T^{80} - 179789984 p^{37} T^{84} + 5390702 p^{40} T^{88} + 3644 p^{44} T^{92} + p^{48} T^{96} \)
59 \( ( 1 - 468 T^{2} + 110536 T^{4} - 17837620 T^{6} + 2243479819 T^{8} - 237445431534 T^{10} + 22184461155196 T^{12} - 1880938370046468 T^{14} + 146878189757422760 T^{16} - 10660350488219692422 T^{18} + \)\(72\!\cdots\!60\)\( T^{20} - \)\(46\!\cdots\!38\)\( T^{22} + \)\(28\!\cdots\!77\)\( T^{24} - \)\(46\!\cdots\!38\)\( p^{2} T^{26} + \)\(72\!\cdots\!60\)\( p^{4} T^{28} - 10660350488219692422 p^{6} T^{30} + 146878189757422760 p^{8} T^{32} - 1880938370046468 p^{10} T^{34} + 22184461155196 p^{12} T^{36} - 237445431534 p^{14} T^{38} + 2243479819 p^{16} T^{40} - 17837620 p^{18} T^{42} + 110536 p^{20} T^{44} - 468 p^{22} T^{46} + p^{24} T^{48} )^{2} \)
61 \( ( 1 + 2 T - 205 T^{2} - 282 T^{3} + 19874 T^{4} + 20324 T^{5} - 1389787 T^{6} - 2654184 T^{7} + 88843230 T^{8} + 244346906 T^{9} - 5628653299 T^{10} - 7579305102 T^{11} + 351718201024 T^{12} - 7579305102 p T^{13} - 5628653299 p^{2} T^{14} + 244346906 p^{3} T^{15} + 88843230 p^{4} T^{16} - 2654184 p^{5} T^{17} - 1389787 p^{6} T^{18} + 20324 p^{7} T^{19} + 19874 p^{8} T^{20} - 282 p^{9} T^{21} - 205 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} )^{4} \)
67 \( ( 1 - 6 T + 18 T^{2} - 512 T^{3} + 3155 T^{4} - 31062 T^{5} + 260654 T^{6} - 112776 T^{7} + 14456827 T^{8} + 105561502 T^{9} - 1175790034 T^{10} - 256514382 T^{11} + 119764811148 T^{12} - 988070527382 T^{13} + 572787537588 T^{14} - 66618207685578 T^{15} + 1065182258006513 T^{16} - 6983349967589020 T^{17} + 77101203579966126 T^{18} - 377038438616132710 T^{19} + 3698754586263283589 T^{20} - 2553520252374996392 T^{21} - \)\(12\!\cdots\!40\)\( T^{22} - \)\(25\!\cdots\!26\)\( T^{23} + \)\(10\!\cdots\!62\)\( T^{24} - \)\(25\!\cdots\!26\)\( p T^{25} - \)\(12\!\cdots\!40\)\( p^{2} T^{26} - 2553520252374996392 p^{3} T^{27} + 3698754586263283589 p^{4} T^{28} - 377038438616132710 p^{5} T^{29} + 77101203579966126 p^{6} T^{30} - 6983349967589020 p^{7} T^{31} + 1065182258006513 p^{8} T^{32} - 66618207685578 p^{9} T^{33} + 572787537588 p^{10} T^{34} - 988070527382 p^{11} T^{35} + 119764811148 p^{12} T^{36} - 256514382 p^{13} T^{37} - 1175790034 p^{14} T^{38} + 105561502 p^{15} T^{39} + 14456827 p^{16} T^{40} - 112776 p^{17} T^{41} + 260654 p^{18} T^{42} - 31062 p^{19} T^{43} + 3155 p^{20} T^{44} - 512 p^{21} T^{45} + 18 p^{22} T^{46} - 6 p^{23} T^{47} + p^{24} T^{48} )^{2} \)
71 \( ( 1 - 380 T^{2} + 67624 T^{4} - 7774462 T^{6} + 695515741 T^{8} - 55208037926 T^{10} + 4070733337884 T^{12} - 55208037926 p^{2} T^{14} + 695515741 p^{4} T^{16} - 7774462 p^{6} T^{18} + 67624 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} )^{4} \)
73 \( ( 1 - 26 T + 338 T^{2} - 2824 T^{3} - 4052 T^{4} + 6532 p T^{5} - 7040672 T^{6} + 64302216 T^{7} - 231632700 T^{8} - 3108405420 T^{9} + 62678418040 T^{10} - 660358771686 T^{11} + 3883606529204 T^{12} + 3154961749866 T^{13} - 321523871998316 T^{14} + 59461546747680 p T^{15} - 33666359593075984 T^{16} + 93517442065036964 T^{17} + 1121374579875446824 T^{18} - 22846284968810985360 T^{19} + \)\(22\!\cdots\!36\)\( T^{20} - \)\(10\!\cdots\!36\)\( T^{21} - \)\(20\!\cdots\!30\)\( T^{22} + \)\(10\!\cdots\!90\)\( T^{23} - \)\(12\!\cdots\!54\)\( T^{24} + \)\(10\!\cdots\!90\)\( p T^{25} - \)\(20\!\cdots\!30\)\( p^{2} T^{26} - \)\(10\!\cdots\!36\)\( p^{3} T^{27} + \)\(22\!\cdots\!36\)\( p^{4} T^{28} - 22846284968810985360 p^{5} T^{29} + 1121374579875446824 p^{6} T^{30} + 93517442065036964 p^{7} T^{31} - 33666359593075984 p^{8} T^{32} + 59461546747680 p^{10} T^{33} - 321523871998316 p^{10} T^{34} + 3154961749866 p^{11} T^{35} + 3883606529204 p^{12} T^{36} - 660358771686 p^{13} T^{37} + 62678418040 p^{14} T^{38} - 3108405420 p^{15} T^{39} - 231632700 p^{16} T^{40} + 64302216 p^{17} T^{41} - 7040672 p^{18} T^{42} + 6532 p^{20} T^{43} - 4052 p^{20} T^{44} - 2824 p^{21} T^{45} + 338 p^{22} T^{46} - 26 p^{23} T^{47} + p^{24} T^{48} )^{2} \)
79 \( ( 1 + 792 T^{2} + 329060 T^{4} + 95389952 T^{6} + 21589636608 T^{8} + 4041079663876 T^{10} + 647104949733444 T^{12} + 90606325864008084 T^{14} + 11257557821030507000 T^{16} + \)\(12\!\cdots\!64\)\( T^{18} + \)\(12\!\cdots\!92\)\( T^{20} + \)\(11\!\cdots\!28\)\( T^{22} + \)\(95\!\cdots\!58\)\( T^{24} + \)\(11\!\cdots\!28\)\( p^{2} T^{26} + \)\(12\!\cdots\!92\)\( p^{4} T^{28} + \)\(12\!\cdots\!64\)\( p^{6} T^{30} + 11257557821030507000 p^{8} T^{32} + 90606325864008084 p^{10} T^{34} + 647104949733444 p^{12} T^{36} + 4041079663876 p^{14} T^{38} + 21589636608 p^{16} T^{40} + 95389952 p^{18} T^{42} + 329060 p^{20} T^{44} + 792 p^{22} T^{46} + p^{24} T^{48} )^{2} \)
83 \( ( 1 + 13034 T^{4} + 41246923 T^{8} - 205524486682 T^{12} - 1648981893231557 T^{16} - 20951379791926319348 T^{20} - \)\(21\!\cdots\!02\)\( T^{24} - 20951379791926319348 p^{4} T^{28} - 1648981893231557 p^{8} T^{32} - 205524486682 p^{12} T^{36} + 41246923 p^{16} T^{40} + 13034 p^{20} T^{44} + p^{24} T^{48} )^{2} \)
89 \( ( 1 - 596 T^{2} + 176335 T^{4} - 33862708 T^{6} + 4732762942 T^{8} - 527180748746 T^{10} + 52996043610893 T^{12} - 5527761205782670 T^{14} + 609010338851795514 T^{16} - 63418967194434678788 T^{18} + 64770036151787787549 p T^{20} - \)\(47\!\cdots\!12\)\( T^{22} + \)\(39\!\cdots\!72\)\( T^{24} - \)\(47\!\cdots\!12\)\( p^{2} T^{26} + 64770036151787787549 p^{5} T^{28} - 63418967194434678788 p^{6} T^{30} + 609010338851795514 p^{8} T^{32} - 5527761205782670 p^{10} T^{34} + 52996043610893 p^{12} T^{36} - 527180748746 p^{14} T^{38} + 4732762942 p^{16} T^{40} - 33862708 p^{18} T^{42} + 176335 p^{20} T^{44} - 596 p^{22} T^{46} + p^{24} T^{48} )^{2} \)
97 \( ( 1 + 30 T + 450 T^{2} + 5578 T^{3} + 63738 T^{4} + 656126 T^{5} + 6558722 T^{6} + 66457714 T^{7} + 746627827 T^{8} + 8806531984 T^{9} + 96125407164 T^{10} + 1008376671640 T^{11} + 10260889235300 T^{12} + 1008376671640 p T^{13} + 96125407164 p^{2} T^{14} + 8806531984 p^{3} T^{15} + 746627827 p^{4} T^{16} + 66457714 p^{5} T^{17} + 6558722 p^{6} T^{18} + 656126 p^{7} T^{19} + 63738 p^{8} T^{20} + 5578 p^{9} T^{21} + 450 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{96} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.36346477388012276580321747064, −2.28310621613916264379906146969, −2.24905959127290116076869027754, −2.22442289446052811584306463215, −2.18424534059329627715729261420, −2.13362566073489665501973051282, −2.12400642151089781795677030840, −2.04814462909678929295636779610, −2.04160873205741505505109639699, −2.03146238300716129021761020825, −1.94251051800903210010206832643, −1.82756279041259597197516985663, −1.74454804715826243937081266219, −1.74408700952844118642915398671, −1.73904343822875846850639853300, −1.62700938985477079762534734678, −1.40907799761246195810057222594, −1.38870816652073950795454864894, −1.24324151723839757288615306673, −1.20366283258916783520851124833, −1.04329792440247530466316119725, −1.00705138499228596445757631019, −0.73766198130993833711533773375, −0.65030445642386813680062537777, −0.33149081114965854031648209193, 0.33149081114965854031648209193, 0.65030445642386813680062537777, 0.73766198130993833711533773375, 1.00705138499228596445757631019, 1.04329792440247530466316119725, 1.20366283258916783520851124833, 1.24324151723839757288615306673, 1.38870816652073950795454864894, 1.40907799761246195810057222594, 1.62700938985477079762534734678, 1.73904343822875846850639853300, 1.74408700952844118642915398671, 1.74454804715826243937081266219, 1.82756279041259597197516985663, 1.94251051800903210010206832643, 2.03146238300716129021761020825, 2.04160873205741505505109639699, 2.04814462909678929295636779610, 2.12400642151089781795677030840, 2.13362566073489665501973051282, 2.18424534059329627715729261420, 2.22442289446052811584306463215, 2.24905959127290116076869027754, 2.28310621613916264379906146969, 2.36346477388012276580321747064

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

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