Properties

Degree 32
Conductor $ 2^{64} \cdot 3^{16} $
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  + 6·4-s − 4·8-s + 32·11-s + 26·16-s − 32·19-s − 128·23-s + 32·29-s + 8·32-s − 96·37-s + 160·43-s + 192·44-s − 336·49-s − 160·53-s − 128·59-s − 32·61-s + 24·64-s + 320·67-s + 512·71-s − 192·76-s − 36·81-s − 160·83-s − 128·88-s − 768·92-s + 384·103-s − 512·109-s − 224·113-s + 192·116-s + ⋯
L(s)  = 1  + 3/2·4-s − 1/2·8-s + 2.90·11-s + 13/8·16-s − 1.68·19-s − 5.56·23-s + 1.10·29-s + 1/4·32-s − 2.59·37-s + 3.72·43-s + 4.36·44-s − 6.85·49-s − 3.01·53-s − 2.16·59-s − 0.524·61-s + 3/8·64-s + 4.77·67-s + 7.21·71-s − 2.52·76-s − 4/9·81-s − 1.92·83-s − 1.45·88-s − 8.34·92-s + 3.72·103-s − 4.69·109-s − 1.98·113-s + 1.65·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{64} \cdot 3^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{48} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.26399\)
\(L(\frac12)\)  \(\approx\)  \(2.26399\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 - 3 p T^{2} + p^{2} T^{3} + 5 p T^{4} - 7 p^{3} T^{5} + 11 p^{3} T^{6} + p^{7} T^{7} - 31 p^{4} T^{8} + p^{9} T^{9} + 11 p^{7} T^{10} - 7 p^{9} T^{11} + 5 p^{9} T^{12} + p^{12} T^{13} - 3 p^{13} T^{14} + p^{16} T^{16} \)
3 \( ( 1 + p^{2} T^{4} )^{4} \)
good5 \( 1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 5824 p^{2} T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 13456544 p T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 75471830656 p^{2} T^{17} + 2740220928 p^{4} T^{18} + 1881828576 p^{6} T^{19} + 753060504 p^{8} T^{20} - 13456544 p^{11} T^{21} + 20875776 p^{12} T^{22} - 2255168 p^{14} T^{23} + 223452 p^{16} T^{24} + 5824 p^{20} T^{25} + 512 p^{20} T^{26} + 5664 p^{22} T^{27} - 344 p^{24} T^{28} + 32 p^{26} T^{29} + p^{32} T^{32} \)
7 \( ( 1 + 24 p T^{2} - 64 p T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 3649664 p^{2} T^{9} + 1070392 p^{4} T^{10} - 56512 p^{6} T^{11} + 15076 p^{8} T^{12} - 64 p^{11} T^{13} + 24 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
11 \( 1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 39251678022944 p T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 61910274521995104 p^{2} T^{17} + 5293166227138048 p^{4} T^{18} - 39251678022944 p^{7} T^{19} + 33797606438392 p^{8} T^{20} - 2916953293728 p^{10} T^{21} + 253620152832 p^{12} T^{22} - 21867108000 p^{14} T^{23} + 2263228700 p^{16} T^{24} - 219655136 p^{18} T^{25} + 18165248 p^{20} T^{26} - 1636576 p^{22} T^{27} + 137032 p^{24} T^{28} - 8480 p^{26} T^{29} + 512 p^{28} T^{30} - 32 p^{30} T^{31} + p^{32} T^{32} \)
13 \( 1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 70586941744778752 p^{2} T^{17} + 7008632866619392 p^{4} T^{18} + 669839496880000 p^{6} T^{19} + 5354592144136 p^{8} T^{20} + 4889456013184 p^{10} T^{21} + 184268595200 p^{12} T^{22} + 7270355200 p^{14} T^{23} + 2004669468 p^{16} T^{24} + 68335872 p^{18} T^{25} + 5120000 p^{20} T^{26} + 95360 p^{22} T^{27} + 7608 p^{24} T^{28} + 3200 p^{26} T^{29} + p^{32} T^{32} \)
17 \( ( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 188800 p T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 1543904000 p^{2} T^{9} + 201700088 p^{4} T^{10} - 188800 p^{7} T^{11} + 516540 p^{8} T^{12} - 2944 p^{10} T^{13} + 968 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( 1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} - \)\(19\!\cdots\!42\)\( T^{16} - 30474951661580761248 p^{2} T^{17} + 3161998119961945600 p^{4} T^{18} + 253693358084547040 p^{6} T^{19} + 998616856187896 p^{8} T^{20} - 698985761087712 p^{10} T^{21} - 30601835632128 p^{12} T^{22} - 703649376 p^{14} T^{23} + 107916937244 p^{16} T^{24} + 4454446304 p^{18} T^{25} + 1945088 p^{20} T^{26} - 8424608 p^{22} T^{27} - 523448 p^{24} T^{28} - 2656 p^{26} T^{29} + 512 p^{28} T^{30} + 32 p^{30} T^{31} + p^{32} T^{32} \)
23 \( ( 1 + 64 T + 152 p T^{2} + 127936 T^{3} + 4410332 T^{4} + 130001728 T^{5} + 3673719192 T^{6} + 94049622208 T^{7} + 2261818535238 T^{8} + 94049622208 p^{2} T^{9} + 3673719192 p^{4} T^{10} + 130001728 p^{6} T^{11} + 4410332 p^{8} T^{12} + 127936 p^{10} T^{13} + 152 p^{13} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
29 \( 1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} + \)\(19\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!40\)\( T^{15} - \)\(82\!\cdots\!10\)\( T^{16} + \)\(11\!\cdots\!40\)\( p^{2} T^{17} + \)\(19\!\cdots\!28\)\( p^{4} T^{18} - 20658732330776531008 p^{6} T^{19} + 694517140133881240 p^{8} T^{20} + 2649303176415616 p^{10} T^{21} - 347423504794624 p^{12} T^{22} + 9791032230816 p^{14} T^{23} + 491466517980 p^{16} T^{24} - 25469097376 p^{18} T^{25} + 304715776 p^{20} T^{26} + 20596992 p^{22} T^{27} - 1552984 p^{24} T^{28} + 18368 p^{26} T^{29} + 512 p^{28} T^{30} - 32 p^{30} T^{31} + p^{32} T^{32} \)
31 \( 1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} + 13661536407528780744 p T^{12} - \)\(51\!\cdots\!00\)\( T^{14} + \)\(53\!\cdots\!38\)\( T^{16} - \)\(51\!\cdots\!00\)\( p^{4} T^{18} + 13661536407528780744 p^{9} T^{20} - 295006946315669072 p^{12} T^{22} + 171125889681052 p^{16} T^{24} - 80335806576 p^{20} T^{26} + 29025544 p^{24} T^{28} - 7312 p^{28} T^{30} + p^{32} T^{32} \)
37 \( 1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 22032204217953568 p T^{11} + 35433653736114978312 T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(43\!\cdots\!90\)\( T^{16} + \)\(14\!\cdots\!84\)\( p^{2} T^{17} + \)\(50\!\cdots\!52\)\( p^{4} T^{18} + \)\(14\!\cdots\!40\)\( p^{6} T^{19} + 35433653736114978312 p^{8} T^{20} + 22032204217953568 p^{11} T^{21} + 21703692469355008 p^{12} T^{22} + 655265530977504 p^{14} T^{23} + 21839639792924 p^{16} T^{24} + 602883756192 p^{18} T^{25} + 12932982272 p^{20} T^{26} + 217733344 p^{22} T^{27} + 4040888 p^{24} T^{28} + 145952 p^{26} T^{29} + 4608 p^{28} T^{30} + 96 p^{30} T^{31} + p^{32} T^{32} \)
41 \( 1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(35\!\cdots\!72\)\( T^{14} + \)\(64\!\cdots\!06\)\( T^{16} - \)\(35\!\cdots\!72\)\( p^{4} T^{18} + \)\(16\!\cdots\!68\)\( p^{8} T^{20} - 6376104819902485008 p^{12} T^{22} + 2044068651261084 p^{16} T^{24} - 524939980080 p^{20} T^{26} + 102706104 p^{24} T^{28} - 13840 p^{28} T^{30} + p^{32} T^{32} \)
43 \( 1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 217203197196384 p T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} - \)\(12\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!84\)\( T^{13} - \)\(44\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!76\)\( T^{15} - \)\(17\!\cdots\!30\)\( T^{16} + \)\(28\!\cdots\!76\)\( p^{2} T^{17} - \)\(44\!\cdots\!20\)\( p^{4} T^{18} + \)\(13\!\cdots\!84\)\( p^{6} T^{19} - \)\(12\!\cdots\!76\)\( p^{8} T^{20} - 7137460469658328480 p^{10} T^{21} + 288453906337733120 p^{12} T^{22} - 217203197196384 p^{15} T^{23} + 336417491247900 p^{16} T^{24} - 8719368905312 p^{18} T^{25} + 178619596288 p^{20} T^{26} - 3813053664 p^{22} T^{27} + 71106632 p^{24} T^{28} - 978464 p^{26} T^{29} + 12800 p^{28} T^{30} - 160 p^{30} T^{31} + p^{32} T^{32} \)
47 \( 1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{16} - \)\(44\!\cdots\!04\)\( p^{4} T^{18} + \)\(15\!\cdots\!16\)\( p^{8} T^{20} - 47271044690493269328 p^{12} T^{22} + 11327375509374492 p^{16} T^{24} - 2097883923184 p^{20} T^{26} + 280869112 p^{24} T^{28} - 24144 p^{28} T^{30} + p^{32} T^{32} \)
53 \( 1 + 160 T + 12800 T^{2} + 602944 T^{3} + 74504 p T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} + \)\(13\!\cdots\!12\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} - \)\(18\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!08\)\( T^{15} + \)\(51\!\cdots\!54\)\( T^{16} + \)\(16\!\cdots\!08\)\( p^{2} T^{17} - \)\(18\!\cdots\!48\)\( p^{4} T^{18} - \)\(17\!\cdots\!00\)\( p^{6} T^{19} + \)\(13\!\cdots\!12\)\( p^{8} T^{20} + 54880211634179791488 p^{10} T^{21} + 1098860393015073792 p^{12} T^{22} + 14471440004155872 p^{14} T^{23} + 34861972067036 p^{16} T^{24} - 3791430241760 p^{18} T^{25} - 105845942272 p^{20} T^{26} - 1481707264 p^{22} T^{27} + 74504 p^{25} T^{28} + 602944 p^{26} T^{29} + 12800 p^{28} T^{30} + 160 p^{30} T^{31} + p^{32} T^{32} \)
59 \( 1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} + \)\(69\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!56\)\( T^{14} + \)\(29\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!38\)\( T^{16} + \)\(29\!\cdots\!24\)\( p^{2} T^{17} + \)\(51\!\cdots\!56\)\( p^{4} T^{18} + \)\(69\!\cdots\!16\)\( p^{6} T^{19} + \)\(10\!\cdots\!40\)\( p^{8} T^{20} + \)\(19\!\cdots\!80\)\( p^{10} T^{21} + 25502997282495045632 p^{12} T^{22} + 316007889653226112 p^{14} T^{23} + 5234572115355804 p^{16} T^{24} + 71627082366848 p^{18} T^{25} + 700645040128 p^{20} T^{26} + 9279937408 p^{22} T^{27} + 136226184 p^{24} T^{28} + 1121408 p^{26} T^{29} + 8192 p^{28} T^{30} + 128 p^{30} T^{31} + p^{32} T^{32} \)
61 \( 1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} - \)\(10\!\cdots\!56\)\( T^{11} - \)\(45\!\cdots\!20\)\( T^{12} - \)\(71\!\cdots\!16\)\( T^{13} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!64\)\( T^{15} + \)\(72\!\cdots\!42\)\( T^{16} + \)\(23\!\cdots\!64\)\( p^{2} T^{17} - \)\(18\!\cdots\!28\)\( p^{4} T^{18} - \)\(71\!\cdots\!16\)\( p^{6} T^{19} - \)\(45\!\cdots\!20\)\( p^{8} T^{20} - \)\(10\!\cdots\!56\)\( p^{10} T^{21} + 275088896591278592 p^{12} T^{22} + 40394929489472928 p^{14} T^{23} + 2018049968078364 p^{16} T^{24} + 2981900088544 p^{18} T^{25} - 16996289024 p^{20} T^{26} - 1520787552 p^{22} T^{27} - 60439624 p^{24} T^{28} - 38048 p^{26} T^{29} + 512 p^{28} T^{30} + 32 p^{30} T^{31} + p^{32} T^{32} \)
67 \( 1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 7837468747328 p T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} + \)\(26\!\cdots\!12\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{13} + \)\(71\!\cdots\!04\)\( T^{14} - \)\(48\!\cdots\!52\)\( T^{15} + \)\(32\!\cdots\!74\)\( T^{16} - \)\(48\!\cdots\!52\)\( p^{2} T^{17} + \)\(71\!\cdots\!04\)\( p^{4} T^{18} - \)\(10\!\cdots\!88\)\( p^{6} T^{19} + \)\(14\!\cdots\!20\)\( p^{8} T^{20} - \)\(19\!\cdots\!80\)\( p^{10} T^{21} + \)\(26\!\cdots\!12\)\( p^{12} T^{22} - 3502096836597496384 p^{14} T^{23} + 43992629224199580 p^{16} T^{24} - 7837468747328 p^{19} T^{25} + 5982745065472 p^{20} T^{26} - 64228593856 p^{22} T^{27} + 641735304 p^{24} T^{28} - 6047552 p^{26} T^{29} + 51200 p^{28} T^{30} - 320 p^{30} T^{31} + p^{32} T^{32} \)
71 \( ( 1 - 256 T + 68104 T^{2} - 10692864 T^{3} + 1610923548 T^{4} - 179723087616 T^{5} + 18972832358712 T^{6} - 1588998739085056 T^{7} + 125568612540426694 T^{8} - 1588998739085056 p^{2} T^{9} + 18972832358712 p^{4} T^{10} - 179723087616 p^{6} T^{11} + 1610923548 p^{8} T^{12} - 10692864 p^{10} T^{13} + 68104 p^{12} T^{14} - 256 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
73 \( 1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} - \)\(15\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(83\!\cdots\!96\)\( T^{14} + \)\(47\!\cdots\!22\)\( T^{16} - \)\(83\!\cdots\!96\)\( p^{4} T^{18} + \)\(12\!\cdots\!76\)\( p^{8} T^{20} - \)\(15\!\cdots\!24\)\( p^{12} T^{22} + 167549428359087132 p^{16} T^{24} - 14391245893936 p^{20} T^{26} + 946714744 p^{24} T^{28} - 42768 p^{28} T^{30} + p^{32} T^{32} \)
79 \( 1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} - \)\(60\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!96\)\( T^{12} - \)\(43\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!62\)\( T^{16} - \)\(43\!\cdots\!36\)\( p^{4} T^{18} + \)\(55\!\cdots\!96\)\( p^{8} T^{20} - \)\(60\!\cdots\!08\)\( p^{12} T^{22} + 534425714020543644 p^{16} T^{24} - 37296559235888 p^{20} T^{26} + 1905826568 p^{24} T^{28} - 62928 p^{28} T^{30} + p^{32} T^{32} \)
83 \( 1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} + \)\(11\!\cdots\!96\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} + \)\(12\!\cdots\!44\)\( T^{13} + \)\(89\!\cdots\!76\)\( T^{14} + \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!66\)\( T^{16} + \)\(74\!\cdots\!76\)\( p^{2} T^{17} + \)\(89\!\cdots\!76\)\( p^{4} T^{18} + \)\(12\!\cdots\!44\)\( p^{6} T^{19} + \)\(13\!\cdots\!00\)\( p^{8} T^{20} + \)\(11\!\cdots\!36\)\( p^{10} T^{21} + \)\(11\!\cdots\!96\)\( p^{12} T^{22} + 1483531366054758688 p^{14} T^{23} + 18280323695716892 p^{16} T^{24} + 174484645067104 p^{18} T^{25} + 1654826188288 p^{20} T^{26} + 16432771168 p^{22} T^{27} + 107479624 p^{24} T^{28} + 895904 p^{26} T^{29} + 12800 p^{28} T^{30} + 160 p^{30} T^{31} + p^{32} T^{32} \)
89 \( 1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} - \)\(23\!\cdots\!04\)\( T^{10} + \)\(28\!\cdots\!72\)\( T^{12} - \)\(29\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!10\)\( T^{16} - \)\(29\!\cdots\!36\)\( p^{4} T^{18} + \)\(28\!\cdots\!72\)\( p^{8} T^{20} - \)\(23\!\cdots\!04\)\( p^{12} T^{22} + 1567286911309649436 p^{16} T^{24} - 82544801381712 p^{20} T^{26} + 3201135736 p^{24} T^{28} - 81008 p^{28} T^{30} + p^{32} T^{32} \)
97 \( ( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 49274039499776 p^{2} T^{9} + 10857255215864 p^{4} T^{10} + 3485408768 p^{6} T^{11} + 770481564 p^{8} T^{12} + 116224 p^{10} T^{13} + 38216 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.76261107057091634498132853322, −4.65808041655909513948753146786, −4.63841283494658027264344300370, −4.49102081397836036664684980251, −4.31816049767233815026115499319, −4.19890698947182026530310114144, −4.18875414088002136895591747798, −4.10867673311570854249237111966, −3.85987524989755973398501719249, −3.54826847181597847311203393806, −3.54790274433153807732361457807, −3.54736934419977414921133792220, −3.52608016002707217453223146512, −3.22780090768164610083624752412, −3.19387372975425808843308269309, −2.80832534364437408566934433913, −2.57643030537763233134342356400, −2.55531396308722075764088651644, −2.24667172879860080816651333436, −2.09422229771337631941299255751, −1.83178116133491254186155661668, −1.74188252212475400268684983584, −1.64318431323729946503727064285, −1.38414053555612297403574408004, −0.49614870507904859657659736481, 0.49614870507904859657659736481, 1.38414053555612297403574408004, 1.64318431323729946503727064285, 1.74188252212475400268684983584, 1.83178116133491254186155661668, 2.09422229771337631941299255751, 2.24667172879860080816651333436, 2.55531396308722075764088651644, 2.57643030537763233134342356400, 2.80832534364437408566934433913, 3.19387372975425808843308269309, 3.22780090768164610083624752412, 3.52608016002707217453223146512, 3.54736934419977414921133792220, 3.54790274433153807732361457807, 3.54826847181597847311203393806, 3.85987524989755973398501719249, 4.10867673311570854249237111966, 4.18875414088002136895591747798, 4.19890698947182026530310114144, 4.31816049767233815026115499319, 4.49102081397836036664684980251, 4.63841283494658027264344300370, 4.65808041655909513948753146786, 4.76261107057091634498132853322

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

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