Properties

Degree 56
Conductor $ 2^{140} \cdot 7^{28} $
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  + 4·7-s + 47·9-s − 6·17-s − 30·23-s + 159·25-s + 6·31-s + 294·47-s − 2·49-s + 188·63-s + 136·71-s + 234·73-s + 162·79-s + 1.17e3·81-s − 150·89-s − 570·103-s + 56·113-s − 24·119-s − 889·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 282·153-s + 157-s − 120·161-s + ⋯
L(s)  = 1  + 4/7·7-s + 47/9·9-s − 0.352·17-s − 1.30·23-s + 6.35·25-s + 6/31·31-s + 6.25·47-s − 0.0408·49-s + 2.98·63-s + 1.91·71-s + 3.20·73-s + 2.05·79-s + 14.4·81-s − 1.68·89-s − 5.53·103-s + 0.495·113-s − 0.201·119-s − 7.34·121-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 − 1.84·153-s + 0.00636·157-s − 0.745·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s+1)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(56\)
\( N \)  =  \(2^{140} \cdot 7^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((56,\ 2^{140} \cdot 7^{28} ,\ ( \ : [1]^{28} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.340903\)
\(L(\frac12)\)  \(\approx\)  \(0.340903\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 56. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 55.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 - 2 T + p T^{2} - 572 T^{3} - 2843 T^{4} + 718 p T^{5} + 723 p^{2} T^{6} + 984 p^{4} T^{7} + 723 p^{4} T^{8} + 718 p^{5} T^{9} - 2843 p^{6} T^{10} - 572 p^{8} T^{11} + p^{11} T^{12} - 2 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
good3 \( 1 - 47 T^{2} + 1037 T^{4} - 14434 T^{6} + 49508 p T^{8} - 155488 p^{2} T^{10} + 564811 p^{3} T^{12} - 6618899 p^{3} T^{14} + 22494476 p^{4} T^{16} - 15238127227 T^{18} + 115218208787 T^{20} - 938337670640 T^{22} + 8478568266823 T^{24} - 8527746099131 p^{2} T^{26} + 313643240650 p^{7} T^{28} - 8527746099131 p^{6} T^{30} + 8478568266823 p^{8} T^{32} - 938337670640 p^{12} T^{34} + 115218208787 p^{16} T^{36} - 15238127227 p^{20} T^{38} + 22494476 p^{28} T^{40} - 6618899 p^{31} T^{42} + 564811 p^{35} T^{44} - 155488 p^{38} T^{46} + 49508 p^{41} T^{48} - 14434 p^{44} T^{50} + 1037 p^{48} T^{52} - 47 p^{52} T^{54} + p^{56} T^{56} \)
5 \( 1 - 159 T^{2} + 11941 T^{4} - 613786 T^{6} + 25953748 T^{8} - 933259816 T^{10} + 27772483393 T^{12} - 137637912461 p T^{14} + 2774541094364 p T^{16} - 7022688826979 p^{2} T^{18} - 1383901535951093 T^{20} + 179010627839992936 T^{22} - 7534052787991326241 T^{24} + 48172460877072918737 p T^{26} - \)\(25\!\cdots\!94\)\( p^{2} T^{28} + 48172460877072918737 p^{5} T^{30} - 7534052787991326241 p^{8} T^{32} + 179010627839992936 p^{12} T^{34} - 1383901535951093 p^{16} T^{36} - 7022688826979 p^{22} T^{38} + 2774541094364 p^{25} T^{40} - 137637912461 p^{29} T^{42} + 27772483393 p^{32} T^{44} - 933259816 p^{36} T^{46} + 25953748 p^{40} T^{48} - 613786 p^{44} T^{50} + 11941 p^{48} T^{52} - 159 p^{52} T^{54} + p^{56} T^{56} \)
11 \( 1 + 889 T^{2} + 394197 T^{4} + 113974214 T^{6} + 23808384236 T^{8} + 3781172020968 T^{10} + 42971787873043 p T^{12} + 48873261119654375 T^{14} + 4671529290703136460 T^{16} + \)\(48\!\cdots\!05\)\( T^{18} + \)\(51\!\cdots\!87\)\( T^{20} + \)\(38\!\cdots\!48\)\( p T^{22} + \)\(34\!\cdots\!87\)\( T^{24} - \)\(58\!\cdots\!79\)\( T^{26} - \)\(10\!\cdots\!58\)\( T^{28} - \)\(58\!\cdots\!79\)\( p^{4} T^{30} + \)\(34\!\cdots\!87\)\( p^{8} T^{32} + \)\(38\!\cdots\!48\)\( p^{13} T^{34} + \)\(51\!\cdots\!87\)\( p^{16} T^{36} + \)\(48\!\cdots\!05\)\( p^{20} T^{38} + 4671529290703136460 p^{24} T^{40} + 48873261119654375 p^{28} T^{42} + 42971787873043 p^{33} T^{44} + 3781172020968 p^{36} T^{46} + 23808384236 p^{40} T^{48} + 113974214 p^{44} T^{50} + 394197 p^{48} T^{52} + 889 p^{52} T^{54} + p^{56} T^{56} \)
13 \( ( 1 + 1362 T^{2} + 925259 T^{4} + 418943860 T^{6} + 142215367113 T^{8} + 38417212028174 T^{10} + 8519228873094603 T^{12} + 1573943470735764696 T^{14} + 8519228873094603 p^{4} T^{16} + 38417212028174 p^{8} T^{18} + 142215367113 p^{12} T^{20} + 418943860 p^{16} T^{22} + 925259 p^{20} T^{24} + 1362 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
17 \( ( 1 + 3 T + 1129 T^{2} + 3378 T^{3} + 549136 T^{4} - 46296 T^{5} + 215152205 T^{6} - 768149427 T^{7} + 96505381532 T^{8} - 5045050689 p T^{9} + 37516126355231 T^{10} + 8050978453608 T^{11} + 11542106309206515 T^{12} - 31916966710270689 T^{13} + 3296984279863636374 T^{14} - 31916966710270689 p^{2} T^{15} + 11542106309206515 p^{4} T^{16} + 8050978453608 p^{6} T^{17} + 37516126355231 p^{8} T^{18} - 5045050689 p^{11} T^{19} + 96505381532 p^{12} T^{20} - 768149427 p^{14} T^{21} + 215152205 p^{16} T^{22} - 46296 p^{18} T^{23} + 549136 p^{20} T^{24} + 3378 p^{22} T^{25} + 1129 p^{24} T^{26} + 3 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
19 \( 1 - 2967 T^{2} + 4328965 T^{4} - 4230786058 T^{6} + 3189115613356 T^{8} - 2013825136716232 T^{10} + 1120452242265367009 T^{12} - \)\(56\!\cdots\!97\)\( T^{14} + \)\(26\!\cdots\!00\)\( T^{16} - \)\(11\!\cdots\!99\)\( T^{18} + \)\(51\!\cdots\!27\)\( T^{20} - \)\(21\!\cdots\!36\)\( T^{22} + \)\(85\!\cdots\!79\)\( T^{24} - \)\(32\!\cdots\!99\)\( T^{26} + \)\(12\!\cdots\!66\)\( T^{28} - \)\(32\!\cdots\!99\)\( p^{4} T^{30} + \)\(85\!\cdots\!79\)\( p^{8} T^{32} - \)\(21\!\cdots\!36\)\( p^{12} T^{34} + \)\(51\!\cdots\!27\)\( p^{16} T^{36} - \)\(11\!\cdots\!99\)\( p^{20} T^{38} + \)\(26\!\cdots\!00\)\( p^{24} T^{40} - \)\(56\!\cdots\!97\)\( p^{28} T^{42} + 1120452242265367009 p^{32} T^{44} - 2013825136716232 p^{36} T^{46} + 3189115613356 p^{40} T^{48} - 4230786058 p^{44} T^{50} + 4328965 p^{48} T^{52} - 2967 p^{52} T^{54} + p^{56} T^{56} \)
23 \( ( 1 + 15 T - 1867 T^{2} - 43222 T^{3} + 1634008 T^{4} + 57305716 T^{5} - 778531571 T^{6} - 52108276687 T^{7} - 145245356 p T^{8} + 34672411729187 T^{9} + 392769218239099 T^{10} - 15799404969401192 T^{11} - 405669572329048181 T^{12} + 3298854034496585159 T^{13} + \)\(25\!\cdots\!50\)\( T^{14} + 3298854034496585159 p^{2} T^{15} - 405669572329048181 p^{4} T^{16} - 15799404969401192 p^{6} T^{17} + 392769218239099 p^{8} T^{18} + 34672411729187 p^{10} T^{19} - 145245356 p^{13} T^{20} - 52108276687 p^{14} T^{21} - 778531571 p^{16} T^{22} + 57305716 p^{18} T^{23} + 1634008 p^{20} T^{24} - 43222 p^{22} T^{25} - 1867 p^{24} T^{26} + 15 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
29 \( ( 1 - 6334 T^{2} + 20962315 T^{4} - 47301049132 T^{6} + 80648119157065 T^{8} - 109385609778467554 T^{10} + \)\(12\!\cdots\!91\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(12\!\cdots\!91\)\( p^{4} T^{16} - 109385609778467554 p^{8} T^{18} + 80648119157065 p^{12} T^{20} - 47301049132 p^{16} T^{22} + 20962315 p^{20} T^{24} - 6334 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
31 \( ( 1 - 3 T + 4717 T^{2} - 14142 T^{3} + 11314480 T^{4} - 1512228 p T^{5} + 20058706121 T^{6} - 111379757133 T^{7} + 30110821252304 T^{8} - 6139531372773 p T^{9} + 39437421948242843 T^{10} - 251082929333757768 T^{11} + 45434812624600911375 T^{12} - \)\(27\!\cdots\!75\)\( T^{13} + \)\(46\!\cdots\!14\)\( T^{14} - \)\(27\!\cdots\!75\)\( p^{2} T^{15} + 45434812624600911375 p^{4} T^{16} - 251082929333757768 p^{6} T^{17} + 39437421948242843 p^{8} T^{18} - 6139531372773 p^{11} T^{19} + 30110821252304 p^{12} T^{20} - 111379757133 p^{14} T^{21} + 20058706121 p^{16} T^{22} - 1512228 p^{19} T^{23} + 11314480 p^{20} T^{24} - 14142 p^{22} T^{25} + 4717 p^{24} T^{26} - 3 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
37 \( 1 + 10961 T^{2} + 58372533 T^{4} + 206851010742 T^{6} + 567831204147636 T^{8} + 1329138252329179464 T^{10} + \)\(28\!\cdots\!29\)\( T^{12} + \)\(54\!\cdots\!59\)\( T^{14} + \)\(10\!\cdots\!68\)\( T^{16} + \)\(17\!\cdots\!37\)\( T^{18} + \)\(28\!\cdots\!63\)\( T^{20} + \)\(45\!\cdots\!28\)\( T^{22} + \)\(69\!\cdots\!43\)\( T^{24} + \)\(10\!\cdots\!61\)\( T^{26} + \)\(14\!\cdots\!94\)\( T^{28} + \)\(10\!\cdots\!61\)\( p^{4} T^{30} + \)\(69\!\cdots\!43\)\( p^{8} T^{32} + \)\(45\!\cdots\!28\)\( p^{12} T^{34} + \)\(28\!\cdots\!63\)\( p^{16} T^{36} + \)\(17\!\cdots\!37\)\( p^{20} T^{38} + \)\(10\!\cdots\!68\)\( p^{24} T^{40} + \)\(54\!\cdots\!59\)\( p^{28} T^{42} + \)\(28\!\cdots\!29\)\( p^{32} T^{44} + 1329138252329179464 p^{36} T^{46} + 567831204147636 p^{40} T^{48} + 206851010742 p^{44} T^{50} + 58372533 p^{48} T^{52} + 10961 p^{52} T^{54} + p^{56} T^{56} \)
41 \( ( 1 - 10382 T^{2} + 49151995 T^{4} - 147671783692 T^{6} + 347451156072041 T^{8} - 750907689158837074 T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(27\!\cdots\!44\)\( T^{14} + \)\(15\!\cdots\!39\)\( p^{4} T^{16} - 750907689158837074 p^{8} T^{18} + 347451156072041 p^{12} T^{20} - 147671783692 p^{16} T^{22} + 49151995 p^{20} T^{24} - 10382 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
43 \( ( 1 - 13310 T^{2} + 86379115 T^{4} - 356336944396 T^{6} + 1040115216939209 T^{8} - 2311688031521159074 T^{10} + \)\(43\!\cdots\!31\)\( T^{12} - \)\(77\!\cdots\!68\)\( T^{14} + \)\(43\!\cdots\!31\)\( p^{4} T^{16} - 2311688031521159074 p^{8} T^{18} + 1040115216939209 p^{12} T^{20} - 356336944396 p^{16} T^{22} + 86379115 p^{20} T^{24} - 13310 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
47 \( ( 1 - 147 T + 19093 T^{2} - 1747830 T^{3} + 142337824 T^{4} - 9928134660 T^{5} + 644555667089 T^{6} - 38804185602669 T^{7} + 2246416485329888 T^{8} - 125217985702175499 T^{9} + 6743198524254185267 T^{10} - \)\(35\!\cdots\!04\)\( T^{11} + \)\(17\!\cdots\!67\)\( T^{12} - \)\(87\!\cdots\!83\)\( T^{13} + \)\(41\!\cdots\!38\)\( T^{14} - \)\(87\!\cdots\!83\)\( p^{2} T^{15} + \)\(17\!\cdots\!67\)\( p^{4} T^{16} - \)\(35\!\cdots\!04\)\( p^{6} T^{17} + 6743198524254185267 p^{8} T^{18} - 125217985702175499 p^{10} T^{19} + 2246416485329888 p^{12} T^{20} - 38804185602669 p^{14} T^{21} + 644555667089 p^{16} T^{22} - 9928134660 p^{18} T^{23} + 142337824 p^{20} T^{24} - 1747830 p^{22} T^{25} + 19093 p^{24} T^{26} - 147 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
53 \( 1 + 26449 T^{2} + 352652085 T^{4} + 3207893793206 T^{6} + 22663404917917940 T^{8} + \)\(13\!\cdots\!80\)\( T^{10} + \)\(68\!\cdots\!69\)\( T^{12} + \)\(31\!\cdots\!03\)\( T^{14} + \)\(13\!\cdots\!12\)\( T^{16} + \)\(52\!\cdots\!85\)\( T^{18} + \)\(19\!\cdots\!15\)\( T^{20} + \)\(65\!\cdots\!20\)\( T^{22} + \)\(21\!\cdots\!99\)\( T^{24} + \)\(64\!\cdots\!49\)\( T^{26} + \)\(18\!\cdots\!58\)\( T^{28} + \)\(64\!\cdots\!49\)\( p^{4} T^{30} + \)\(21\!\cdots\!99\)\( p^{8} T^{32} + \)\(65\!\cdots\!20\)\( p^{12} T^{34} + \)\(19\!\cdots\!15\)\( p^{16} T^{36} + \)\(52\!\cdots\!85\)\( p^{20} T^{38} + \)\(13\!\cdots\!12\)\( p^{24} T^{40} + \)\(31\!\cdots\!03\)\( p^{28} T^{42} + \)\(68\!\cdots\!69\)\( p^{32} T^{44} + \)\(13\!\cdots\!80\)\( p^{36} T^{46} + 22663404917917940 p^{40} T^{48} + 3207893793206 p^{44} T^{50} + 352652085 p^{48} T^{52} + 26449 p^{52} T^{54} + p^{56} T^{56} \)
59 \( 1 - 23199 T^{2} + 270681901 T^{4} - 2101260465154 T^{6} + 12140609133783244 T^{8} - 55996258329093534496 T^{10} + \)\(21\!\cdots\!41\)\( T^{12} - \)\(75\!\cdots\!29\)\( T^{14} + \)\(23\!\cdots\!12\)\( T^{16} - \)\(58\!\cdots\!47\)\( T^{18} + \)\(73\!\cdots\!43\)\( T^{20} + \)\(28\!\cdots\!84\)\( T^{22} - \)\(27\!\cdots\!45\)\( T^{24} + \)\(13\!\cdots\!61\)\( T^{26} - \)\(51\!\cdots\!78\)\( T^{28} + \)\(13\!\cdots\!61\)\( p^{4} T^{30} - \)\(27\!\cdots\!45\)\( p^{8} T^{32} + \)\(28\!\cdots\!84\)\( p^{12} T^{34} + \)\(73\!\cdots\!43\)\( p^{16} T^{36} - \)\(58\!\cdots\!47\)\( p^{20} T^{38} + \)\(23\!\cdots\!12\)\( p^{24} T^{40} - \)\(75\!\cdots\!29\)\( p^{28} T^{42} + \)\(21\!\cdots\!41\)\( p^{32} T^{44} - 55996258329093534496 p^{36} T^{46} + 12140609133783244 p^{40} T^{48} - 2101260465154 p^{44} T^{50} + 270681901 p^{48} T^{52} - 23199 p^{52} T^{54} + p^{56} T^{56} \)
61 \( 1 - 29823 T^{2} + 460501717 T^{4} - 4722815853994 T^{6} + 35200887838736692 T^{8} - \)\(19\!\cdots\!04\)\( T^{10} + \)\(78\!\cdots\!85\)\( T^{12} - \)\(18\!\cdots\!29\)\( T^{14} - \)\(94\!\cdots\!76\)\( T^{16} + \)\(33\!\cdots\!45\)\( T^{18} - \)\(16\!\cdots\!05\)\( T^{20} + \)\(28\!\cdots\!28\)\( T^{22} + \)\(14\!\cdots\!11\)\( T^{24} - \)\(14\!\cdots\!83\)\( T^{26} + \)\(68\!\cdots\!94\)\( T^{28} - \)\(14\!\cdots\!83\)\( p^{4} T^{30} + \)\(14\!\cdots\!11\)\( p^{8} T^{32} + \)\(28\!\cdots\!28\)\( p^{12} T^{34} - \)\(16\!\cdots\!05\)\( p^{16} T^{36} + \)\(33\!\cdots\!45\)\( p^{20} T^{38} - \)\(94\!\cdots\!76\)\( p^{24} T^{40} - \)\(18\!\cdots\!29\)\( p^{28} T^{42} + \)\(78\!\cdots\!85\)\( p^{32} T^{44} - \)\(19\!\cdots\!04\)\( p^{36} T^{46} + 35200887838736692 p^{40} T^{48} - 4722815853994 p^{44} T^{50} + 460501717 p^{48} T^{52} - 29823 p^{52} T^{54} + p^{56} T^{56} \)
67 \( 1 + 31217 T^{2} + 517787709 T^{4} + 5841524989710 T^{6} + 49503798181912524 T^{8} + \)\(33\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!61\)\( T^{12} + \)\(90\!\cdots\!79\)\( T^{14} + \)\(41\!\cdots\!04\)\( T^{16} + \)\(19\!\cdots\!45\)\( T^{18} + \)\(94\!\cdots\!95\)\( T^{20} + \)\(44\!\cdots\!04\)\( T^{22} + \)\(19\!\cdots\!67\)\( T^{24} + \)\(83\!\cdots\!85\)\( T^{26} + \)\(36\!\cdots\!94\)\( T^{28} + \)\(83\!\cdots\!85\)\( p^{4} T^{30} + \)\(19\!\cdots\!67\)\( p^{8} T^{32} + \)\(44\!\cdots\!04\)\( p^{12} T^{34} + \)\(94\!\cdots\!95\)\( p^{16} T^{36} + \)\(19\!\cdots\!45\)\( p^{20} T^{38} + \)\(41\!\cdots\!04\)\( p^{24} T^{40} + \)\(90\!\cdots\!79\)\( p^{28} T^{42} + \)\(18\!\cdots\!61\)\( p^{32} T^{44} + \)\(33\!\cdots\!20\)\( p^{36} T^{46} + 49503798181912524 p^{40} T^{48} + 5841524989710 p^{44} T^{50} + 517787709 p^{48} T^{52} + 31217 p^{52} T^{54} + p^{56} T^{56} \)
71 \( ( 1 - 34 T + 28011 T^{2} - 746676 T^{3} + 355428809 T^{4} - 7520211518 T^{5} + 2714440344939 T^{6} - 46583801488728 T^{7} + 2714440344939 p^{2} T^{8} - 7520211518 p^{4} T^{9} + 355428809 p^{6} T^{10} - 746676 p^{8} T^{11} + 28011 p^{10} T^{12} - 34 p^{12} T^{13} + p^{14} T^{14} )^{4} \)
73 \( ( 1 - 117 T + 20089 T^{2} - 1816542 T^{3} + 179447608 T^{4} - 15861607992 T^{5} + 1306352417981 T^{6} - 96234474310155 T^{7} + 6621443801699324 T^{8} - 314069015720954169 T^{9} + 18627105313841417663 T^{10} - \)\(38\!\cdots\!72\)\( T^{11} - \)\(13\!\cdots\!69\)\( T^{12} + \)\(30\!\cdots\!59\)\( T^{13} - \)\(31\!\cdots\!66\)\( T^{14} + \)\(30\!\cdots\!59\)\( p^{2} T^{15} - \)\(13\!\cdots\!69\)\( p^{4} T^{16} - \)\(38\!\cdots\!72\)\( p^{6} T^{17} + 18627105313841417663 p^{8} T^{18} - 314069015720954169 p^{10} T^{19} + 6621443801699324 p^{12} T^{20} - 96234474310155 p^{14} T^{21} + 1306352417981 p^{16} T^{22} - 15861607992 p^{18} T^{23} + 179447608 p^{20} T^{24} - 1816542 p^{22} T^{25} + 20089 p^{24} T^{26} - 117 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
79 \( ( 1 - 81 T - 18635 T^{2} + 1987690 T^{3} + 141701128 T^{4} - 21544064156 T^{5} - 555633261347 T^{6} + 141422275016929 T^{7} + 1190023709785036 T^{8} - 829822752636238525 T^{9} + 11948506914814542235 T^{10} + \)\(48\!\cdots\!28\)\( T^{11} - \)\(31\!\cdots\!05\)\( T^{12} - \)\(14\!\cdots\!61\)\( T^{13} + \)\(29\!\cdots\!22\)\( T^{14} - \)\(14\!\cdots\!61\)\( p^{2} T^{15} - \)\(31\!\cdots\!05\)\( p^{4} T^{16} + \)\(48\!\cdots\!28\)\( p^{6} T^{17} + 11948506914814542235 p^{8} T^{18} - 829822752636238525 p^{10} T^{19} + 1190023709785036 p^{12} T^{20} + 141422275016929 p^{14} T^{21} - 555633261347 p^{16} T^{22} - 21544064156 p^{18} T^{23} + 141701128 p^{20} T^{24} + 1987690 p^{22} T^{25} - 18635 p^{24} T^{26} - 81 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
83 \( ( 1 + 74850 T^{2} + 2696636555 T^{4} + 62023573556788 T^{6} + 1018710990945739401 T^{8} + \)\(12\!\cdots\!26\)\( T^{10} + \)\(12\!\cdots\!87\)\( T^{12} + \)\(94\!\cdots\!44\)\( T^{14} + \)\(12\!\cdots\!87\)\( p^{4} T^{16} + \)\(12\!\cdots\!26\)\( p^{8} T^{18} + 1018710990945739401 p^{12} T^{20} + 62023573556788 p^{16} T^{22} + 2696636555 p^{20} T^{24} + 74850 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
89 \( ( 1 + 75 T + 36841 T^{2} + 2622450 T^{3} + 738257752 T^{4} + 44124301080 T^{5} + 9208981729229 T^{6} + 353090800714677 T^{7} + 69937342527311516 T^{8} - 784957219947982713 T^{9} + \)\(18\!\cdots\!71\)\( T^{10} - \)\(66\!\cdots\!40\)\( p T^{11} - \)\(25\!\cdots\!09\)\( T^{12} - \)\(81\!\cdots\!97\)\( T^{13} - \)\(35\!\cdots\!38\)\( T^{14} - \)\(81\!\cdots\!97\)\( p^{2} T^{15} - \)\(25\!\cdots\!09\)\( p^{4} T^{16} - \)\(66\!\cdots\!40\)\( p^{7} T^{17} + \)\(18\!\cdots\!71\)\( p^{8} T^{18} - 784957219947982713 p^{10} T^{19} + 69937342527311516 p^{12} T^{20} + 353090800714677 p^{14} T^{21} + 9208981729229 p^{16} T^{22} + 44124301080 p^{18} T^{23} + 738257752 p^{20} T^{24} + 2622450 p^{22} T^{25} + 36841 p^{24} T^{26} + 75 p^{26} T^{27} + p^{28} T^{28} )^{2} \)
97 \( ( 1 - 47102 T^{2} + 1120136443 T^{4} - 19507757128876 T^{6} + 288299184325533065 T^{8} - \)\(37\!\cdots\!22\)\( T^{10} + \)\(41\!\cdots\!87\)\( T^{12} - \)\(41\!\cdots\!80\)\( T^{14} + \)\(41\!\cdots\!87\)\( p^{4} T^{16} - \)\(37\!\cdots\!22\)\( p^{8} T^{18} + 288299184325533065 p^{12} T^{20} - 19507757128876 p^{16} T^{22} + 1120136443 p^{20} T^{24} - 47102 p^{24} T^{26} + p^{28} T^{28} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.24675742952826756229698691915, −2.22296676223206936382118416601, −1.95853014266626727549651834080, −1.92519523174617786973092882608, −1.87214536882345445114836127537, −1.80634318410256910302443363107, −1.54109174192404425810509035973, −1.52626448428376685626714303281, −1.50647459479961858795135549538, −1.50390832425996056557446751051, −1.50176122273856486509819863595, −1.44201415134418423082223040576, −1.35557436274588786896393274427, −1.20478158324026177384312385866, −1.11108291014655365970647353594, −0.997564513355930486717655762098, −0.983074735940171440461394743583, −0.926952523146857684531643855841, −0.888892850848509171333454312569, −0.77854643502638731759591847151, −0.58901567391800494428489719855, −0.46612297442672134060979550935, −0.43363270555767420681667953325, −0.35142182324405637853888713464, −0.00619307178390111354362978376, 0.00619307178390111354362978376, 0.35142182324405637853888713464, 0.43363270555767420681667953325, 0.46612297442672134060979550935, 0.58901567391800494428489719855, 0.77854643502638731759591847151, 0.888892850848509171333454312569, 0.926952523146857684531643855841, 0.983074735940171440461394743583, 0.997564513355930486717655762098, 1.11108291014655365970647353594, 1.20478158324026177384312385866, 1.35557436274588786896393274427, 1.44201415134418423082223040576, 1.50176122273856486509819863595, 1.50390832425996056557446751051, 1.50647459479961858795135549538, 1.52626448428376685626714303281, 1.54109174192404425810509035973, 1.80634318410256910302443363107, 1.87214536882345445114836127537, 1.92519523174617786973092882608, 1.95853014266626727549651834080, 2.22296676223206936382118416601, 2.24675742952826756229698691915

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

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