Properties

Degree $32$
Conductor $3.411\times 10^{36}$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·11-s + 32·19-s + 128·23-s + 32·29-s − 96·37-s − 160·43-s − 336·49-s − 160·53-s + 128·59-s − 32·61-s − 320·67-s − 512·71-s − 36·81-s + 160·83-s − 384·103-s − 512·109-s − 224·113-s + 512·121-s − 32·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2.90·11-s + 1.68·19-s + 5.56·23-s + 1.10·29-s − 2.59·37-s − 3.72·43-s − 6.85·49-s − 3.01·53-s + 2.16·59-s − 0.524·61-s − 4.77·67-s − 7.21·71-s − 4/9·81-s + 1.92·83-s − 3.72·103-s − 4.69·109-s − 1.98·113-s + 4.23·121-s − 0.255·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{96} \cdot 3^{16}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{192} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{96} \cdot 3^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00702728\)
\(L(\frac12)\) \(\approx\) \(0.00702728\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.30047537901743377207117223321, −3.26794893808734076888771456868, −3.17116707768128605861325790570, −2.98848440125084608347592057126, −2.91491663454029029810670624769, −2.90546082461030440769227356108, −2.79426399899081875995885310821, −2.76575054861063589596480374997, −2.69574098828840375791067077269, −2.69454776926109546435675362522, −2.31202616218836651569536179449, −2.25064653277094551811092433716, −2.08568911128264950704459460023, −1.86971880102786645922081865275, −1.67881635611071963868465396561, −1.64485216200630904756813040301, −1.54574240647416724279101055534, −1.32521575565122603045165404621, −1.21514445701096959665337310196, −1.20349317546440585800979460819, −1.18613042558188598645488545250, −0.994332729000749753185258906975, −0.12757559482829528313080772147, −0.096982338384529576042743917824, −0.07820144796162669404181266296, 0.07820144796162669404181266296, 0.096982338384529576042743917824, 0.12757559482829528313080772147, 0.994332729000749753185258906975, 1.18613042558188598645488545250, 1.20349317546440585800979460819, 1.21514445701096959665337310196, 1.32521575565122603045165404621, 1.54574240647416724279101055534, 1.64485216200630904756813040301, 1.67881635611071963868465396561, 1.86971880102786645922081865275, 2.08568911128264950704459460023, 2.25064653277094551811092433716, 2.31202616218836651569536179449, 2.69454776926109546435675362522, 2.69574098828840375791067077269, 2.76575054861063589596480374997, 2.79426399899081875995885310821, 2.90546082461030440769227356108, 2.91491663454029029810670624769, 2.98848440125084608347592057126, 3.17116707768128605861325790570, 3.26794893808734076888771456868, 3.30047537901743377207117223321

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

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