Properties

Label 28-74e14-1.1-c4e14-0-0
Degree $28$
Conductor $1.477\times 10^{26}$
Sign $1$
Analytic cond. $2.34836\times 10^{12}$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 28·2-s + 392·4-s − 12·5-s + 48·7-s − 3.58e3·8-s + 394·9-s + 336·10-s − 56·13-s − 1.34e3·14-s + 2.32e4·16-s − 348·17-s − 1.10e4·18-s − 184·19-s − 4.70e3·20-s − 502·23-s + 72·25-s + 1.56e3·26-s + 1.88e4·28-s − 474·29-s − 630·31-s − 1.07e5·32-s + 9.74e3·34-s − 576·35-s + 1.54e5·36-s − 2.54e3·37-s + 5.15e3·38-s + 4.30e4·40-s + ⋯
L(s)  = 1  − 7·2-s + 49/2·4-s − 0.479·5-s + 0.979·7-s − 56·8-s + 4.86·9-s + 3.35·10-s − 0.331·13-s − 6.85·14-s + 91·16-s − 1.20·17-s − 34.0·18-s − 0.509·19-s − 11.7·20-s − 0.948·23-s + 0.115·25-s + 2.31·26-s + 24·28-s − 0.563·29-s − 0.655·31-s − 105·32-s + 8.42·34-s − 0.470·35-s + 119.·36-s − 1.85·37-s + 3.56·38-s + 26.8·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{14} \cdot 37^{14}\)
Sign: $1$
Analytic conductor: \(2.34836\times 10^{12}\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{74} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [2]^{14} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(7.155446577\times10^{-8}\)
\(L(\frac12)\) \(\approx\) \(7.155446577\times10^{-8}\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{2} T + p^{3} T^{2} )^{7} \)
37 \( 1 + 2544 T + 6957359 T^{2} + 355685792 p T^{3} + 18562000697 p^{2} T^{4} + 733759969744 p^{3} T^{5} + 31475360416911 p^{4} T^{6} + 31271431299520 p^{6} T^{7} + 31475360416911 p^{8} T^{8} + 733759969744 p^{11} T^{9} + 18562000697 p^{14} T^{10} + 355685792 p^{17} T^{11} + 6957359 p^{20} T^{12} + 2544 p^{24} T^{13} + p^{28} T^{14} \)
good3 \( 1 - 394 T^{2} + 85051 T^{4} - 4489654 p T^{6} + 190915985 p^{2} T^{8} - 20563844282 p^{2} T^{10} + 216342796100 p^{4} T^{12} - 2048126398358 p^{6} T^{14} + 216342796100 p^{12} T^{16} - 20563844282 p^{18} T^{18} + 190915985 p^{26} T^{20} - 4489654 p^{33} T^{22} + 85051 p^{40} T^{24} - 394 p^{48} T^{26} + p^{56} T^{28} \)
5 \( 1 + 12 T + 72 T^{2} + 2844 p T^{3} - 213098 T^{4} + 2486568 T^{5} + 146286072 T^{6} + 8780919948 T^{7} + 317380465353 T^{8} - 2574111991608 T^{9} + 105388723836216 T^{10} - 210309660295008 T^{11} - 37629164093474656 T^{12} + 2902308242516634828 T^{13} + 42727753081405713168 T^{14} + 2902308242516634828 p^{4} T^{15} - 37629164093474656 p^{8} T^{16} - 210309660295008 p^{12} T^{17} + 105388723836216 p^{16} T^{18} - 2574111991608 p^{20} T^{19} + 317380465353 p^{24} T^{20} + 8780919948 p^{28} T^{21} + 146286072 p^{32} T^{22} + 2486568 p^{36} T^{23} - 213098 p^{40} T^{24} + 2844 p^{45} T^{25} + 72 p^{48} T^{26} + 12 p^{52} T^{27} + p^{56} T^{28} \)
7 \( ( 1 - 24 T + 6726 T^{2} - 51742 T^{3} + 18052794 T^{4} + 587182460 T^{5} + 21407683291 T^{6} + 2651018020764 T^{7} + 21407683291 p^{4} T^{8} + 587182460 p^{8} T^{9} + 18052794 p^{12} T^{10} - 51742 p^{16} T^{11} + 6726 p^{20} T^{12} - 24 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
11 \( 1 - 85002 T^{2} + 4114081755 T^{4} - 140668263843970 T^{6} + 3745236095209650585 T^{8} - \)\(81\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!84\)\( p T^{12} - \)\(23\!\cdots\!26\)\( T^{14} + \)\(13\!\cdots\!84\)\( p^{9} T^{16} - \)\(81\!\cdots\!38\)\( p^{16} T^{18} + 3745236095209650585 p^{24} T^{20} - 140668263843970 p^{32} T^{22} + 4114081755 p^{40} T^{24} - 85002 p^{48} T^{26} + p^{56} T^{28} \)
13 \( 1 + 56 T + 1568 T^{2} - 2724856 T^{3} - 3165213706 T^{4} + 5539540132 T^{5} + 8985689448768 T^{6} + 12875785786747712 T^{7} + 304950936705328717 p T^{8} - \)\(36\!\cdots\!32\)\( T^{9} - \)\(33\!\cdots\!72\)\( T^{10} - \)\(17\!\cdots\!76\)\( p T^{11} - \)\(16\!\cdots\!92\)\( T^{12} + \)\(67\!\cdots\!92\)\( T^{13} + \)\(41\!\cdots\!24\)\( T^{14} + \)\(67\!\cdots\!92\)\( p^{4} T^{15} - \)\(16\!\cdots\!92\)\( p^{8} T^{16} - \)\(17\!\cdots\!76\)\( p^{13} T^{17} - \)\(33\!\cdots\!72\)\( p^{16} T^{18} - \)\(36\!\cdots\!32\)\( p^{20} T^{19} + 304950936705328717 p^{25} T^{20} + 12875785786747712 p^{28} T^{21} + 8985689448768 p^{32} T^{22} + 5539540132 p^{36} T^{23} - 3165213706 p^{40} T^{24} - 2724856 p^{44} T^{25} + 1568 p^{48} T^{26} + 56 p^{52} T^{27} + p^{56} T^{28} \)
17 \( 1 + 348 T + 60552 T^{2} - 9208724 T^{3} - 20637046009 T^{4} - 5605095097528 T^{5} - 658558385148688 T^{6} - 19269798768974168 T^{7} + \)\(12\!\cdots\!65\)\( T^{8} + \)\(49\!\cdots\!68\)\( T^{9} + \)\(71\!\cdots\!48\)\( p T^{10} + \)\(42\!\cdots\!60\)\( T^{11} - \)\(16\!\cdots\!53\)\( T^{12} - \)\(47\!\cdots\!36\)\( T^{13} - \)\(14\!\cdots\!40\)\( T^{14} - \)\(47\!\cdots\!36\)\( p^{4} T^{15} - \)\(16\!\cdots\!53\)\( p^{8} T^{16} + \)\(42\!\cdots\!60\)\( p^{12} T^{17} + \)\(71\!\cdots\!48\)\( p^{17} T^{18} + \)\(49\!\cdots\!68\)\( p^{20} T^{19} + \)\(12\!\cdots\!65\)\( p^{24} T^{20} - 19269798768974168 p^{28} T^{21} - 658558385148688 p^{32} T^{22} - 5605095097528 p^{36} T^{23} - 20637046009 p^{40} T^{24} - 9208724 p^{44} T^{25} + 60552 p^{48} T^{26} + 348 p^{52} T^{27} + p^{56} T^{28} \)
19 \( 1 + 184 T + 16928 T^{2} + 17619296 T^{3} + 15867618359 T^{4} + 2617590392536 T^{5} + 368249384413280 T^{6} + 626089397840613048 T^{7} + \)\(49\!\cdots\!09\)\( T^{8} - \)\(46\!\cdots\!28\)\( T^{9} - \)\(54\!\cdots\!68\)\( T^{10} + \)\(91\!\cdots\!32\)\( T^{11} + \)\(40\!\cdots\!75\)\( T^{12} + \)\(30\!\cdots\!36\)\( T^{13} + \)\(16\!\cdots\!32\)\( T^{14} + \)\(30\!\cdots\!36\)\( p^{4} T^{15} + \)\(40\!\cdots\!75\)\( p^{8} T^{16} + \)\(91\!\cdots\!32\)\( p^{12} T^{17} - \)\(54\!\cdots\!68\)\( p^{16} T^{18} - \)\(46\!\cdots\!28\)\( p^{20} T^{19} + \)\(49\!\cdots\!09\)\( p^{24} T^{20} + 626089397840613048 p^{28} T^{21} + 368249384413280 p^{32} T^{22} + 2617590392536 p^{36} T^{23} + 15867618359 p^{40} T^{24} + 17619296 p^{44} T^{25} + 16928 p^{48} T^{26} + 184 p^{52} T^{27} + p^{56} T^{28} \)
23 \( 1 + 502 T + 126002 T^{2} - 179350290 T^{3} - 257807725322 T^{4} - 85939758964330 T^{5} + 5425793267471034 T^{6} + 37683286712771016358 T^{7} + \)\(23\!\cdots\!21\)\( T^{8} - \)\(42\!\cdots\!72\)\( T^{9} - \)\(49\!\cdots\!08\)\( T^{10} - \)\(40\!\cdots\!40\)\( T^{11} - \)\(73\!\cdots\!00\)\( T^{12} + \)\(87\!\cdots\!92\)\( T^{13} + \)\(60\!\cdots\!36\)\( T^{14} + \)\(87\!\cdots\!92\)\( p^{4} T^{15} - \)\(73\!\cdots\!00\)\( p^{8} T^{16} - \)\(40\!\cdots\!40\)\( p^{12} T^{17} - \)\(49\!\cdots\!08\)\( p^{16} T^{18} - \)\(42\!\cdots\!72\)\( p^{20} T^{19} + \)\(23\!\cdots\!21\)\( p^{24} T^{20} + 37683286712771016358 p^{28} T^{21} + 5425793267471034 p^{32} T^{22} - 85939758964330 p^{36} T^{23} - 257807725322 p^{40} T^{24} - 179350290 p^{44} T^{25} + 126002 p^{48} T^{26} + 502 p^{52} T^{27} + p^{56} T^{28} \)
29 \( 1 + 474 T + 112338 T^{2} + 62620330 T^{3} + 276408573086 T^{4} + 286931828710502 T^{5} + 106915153390097330 T^{6} + \)\(16\!\cdots\!22\)\( T^{7} - \)\(33\!\cdots\!27\)\( T^{8} - \)\(17\!\cdots\!84\)\( T^{9} - \)\(22\!\cdots\!08\)\( T^{10} + \)\(34\!\cdots\!84\)\( T^{11} - \)\(14\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!44\)\( T^{13} - \)\(48\!\cdots\!92\)\( T^{14} - \)\(12\!\cdots\!44\)\( p^{4} T^{15} - \)\(14\!\cdots\!44\)\( p^{8} T^{16} + \)\(34\!\cdots\!84\)\( p^{12} T^{17} - \)\(22\!\cdots\!08\)\( p^{16} T^{18} - \)\(17\!\cdots\!84\)\( p^{20} T^{19} - \)\(33\!\cdots\!27\)\( p^{24} T^{20} + \)\(16\!\cdots\!22\)\( p^{28} T^{21} + 106915153390097330 p^{32} T^{22} + 286931828710502 p^{36} T^{23} + 276408573086 p^{40} T^{24} + 62620330 p^{44} T^{25} + 112338 p^{48} T^{26} + 474 p^{52} T^{27} + p^{56} T^{28} \)
31 \( 1 + 630 T + 198450 T^{2} + 217065406 T^{3} - 1229066036006 T^{4} - 1052289838784874 T^{5} - 395475748348107502 T^{6} - \)\(56\!\cdots\!38\)\( T^{7} - \)\(57\!\cdots\!27\)\( T^{8} - \)\(28\!\cdots\!04\)\( T^{9} - \)\(73\!\cdots\!72\)\( T^{10} - \)\(47\!\cdots\!04\)\( T^{11} + \)\(33\!\cdots\!48\)\( p^{2} T^{12} + \)\(33\!\cdots\!04\)\( p T^{13} + \)\(58\!\cdots\!28\)\( T^{14} + \)\(33\!\cdots\!04\)\( p^{5} T^{15} + \)\(33\!\cdots\!48\)\( p^{10} T^{16} - \)\(47\!\cdots\!04\)\( p^{12} T^{17} - \)\(73\!\cdots\!72\)\( p^{16} T^{18} - \)\(28\!\cdots\!04\)\( p^{20} T^{19} - \)\(57\!\cdots\!27\)\( p^{24} T^{20} - \)\(56\!\cdots\!38\)\( p^{28} T^{21} - 395475748348107502 p^{32} T^{22} - 1052289838784874 p^{36} T^{23} - 1229066036006 p^{40} T^{24} + 217065406 p^{44} T^{25} + 198450 p^{48} T^{26} + 630 p^{52} T^{27} + p^{56} T^{28} \)
41 \( 1 - 19524398 T^{2} + 184948999818539 T^{4} - \)\(11\!\cdots\!14\)\( T^{6} + \)\(48\!\cdots\!01\)\( T^{8} - \)\(16\!\cdots\!94\)\( T^{10} + \)\(46\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!78\)\( T^{14} + \)\(46\!\cdots\!72\)\( p^{8} T^{16} - \)\(16\!\cdots\!94\)\( p^{16} T^{18} + \)\(48\!\cdots\!01\)\( p^{24} T^{20} - \)\(11\!\cdots\!14\)\( p^{32} T^{22} + 184948999818539 p^{40} T^{24} - 19524398 p^{48} T^{26} + p^{56} T^{28} \)
43 \( 1 - 1936 T + 1874048 T^{2} - 15836584 p T^{3} - 19739758413961 T^{4} + 37135213465156088 T^{5} - 34668656302741919968 T^{6} + \)\(71\!\cdots\!24\)\( T^{7} + \)\(83\!\cdots\!77\)\( T^{8} - \)\(28\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!88\)\( T^{10} - \)\(16\!\cdots\!92\)\( T^{11} + \)\(11\!\cdots\!83\)\( T^{12} + \)\(39\!\cdots\!96\)\( T^{13} - \)\(66\!\cdots\!28\)\( T^{14} + \)\(39\!\cdots\!96\)\( p^{4} T^{15} + \)\(11\!\cdots\!83\)\( p^{8} T^{16} - \)\(16\!\cdots\!92\)\( p^{12} T^{17} + \)\(49\!\cdots\!88\)\( p^{16} T^{18} - \)\(28\!\cdots\!68\)\( p^{20} T^{19} + \)\(83\!\cdots\!77\)\( p^{24} T^{20} + \)\(71\!\cdots\!24\)\( p^{28} T^{21} - 34668656302741919968 p^{32} T^{22} + 37135213465156088 p^{36} T^{23} - 19739758413961 p^{40} T^{24} - 15836584 p^{45} T^{25} + 1874048 p^{48} T^{26} - 1936 p^{52} T^{27} + p^{56} T^{28} \)
47 \( ( 1 - 2858 T + 16702730 T^{2} - 38868072582 T^{3} + 156182657105096 T^{4} - 318036430116401934 T^{5} + \)\(10\!\cdots\!77\)\( T^{6} - \)\(18\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!77\)\( p^{4} T^{8} - 318036430116401934 p^{8} T^{9} + 156182657105096 p^{12} T^{10} - 38868072582 p^{16} T^{11} + 16702730 p^{20} T^{12} - 2858 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
53 \( ( 1 + 10114 T + 64251564 T^{2} + 290203277974 T^{3} + 1064908639250588 T^{4} + 3332341116188935666 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(94\!\cdots\!67\)\( p^{4} T^{8} + 3332341116188935666 p^{8} T^{9} + 1064908639250588 p^{12} T^{10} + 290203277974 p^{16} T^{11} + 64251564 p^{20} T^{12} + 10114 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
59 \( 1 + 4502 T + 10134002 T^{2} + 70800952390 T^{3} - 120345836546625 T^{4} - 1468558928300356636 T^{5} - \)\(28\!\cdots\!72\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!65\)\( T^{8} + \)\(31\!\cdots\!46\)\( T^{9} + \)\(65\!\cdots\!10\)\( T^{10} + \)\(41\!\cdots\!70\)\( T^{11} - \)\(31\!\cdots\!05\)\( T^{12} - \)\(64\!\cdots\!72\)\( T^{13} - \)\(12\!\cdots\!60\)\( T^{14} - \)\(64\!\cdots\!72\)\( p^{4} T^{15} - \)\(31\!\cdots\!05\)\( p^{8} T^{16} + \)\(41\!\cdots\!70\)\( p^{12} T^{17} + \)\(65\!\cdots\!10\)\( p^{16} T^{18} + \)\(31\!\cdots\!46\)\( p^{20} T^{19} + \)\(17\!\cdots\!65\)\( p^{24} T^{20} - \)\(18\!\cdots\!60\)\( p^{28} T^{21} - \)\(28\!\cdots\!72\)\( p^{32} T^{22} - 1468558928300356636 p^{36} T^{23} - 120345836546625 p^{40} T^{24} + 70800952390 p^{44} T^{25} + 10134002 p^{48} T^{26} + 4502 p^{52} T^{27} + p^{56} T^{28} \)
61 \( 1 + 11906 T + 70876418 T^{2} + 203665889578 T^{3} - 394142745410454 T^{4} - 5986165856758813458 T^{5} - \)\(22\!\cdots\!34\)\( T^{6} - \)\(63\!\cdots\!30\)\( T^{7} + \)\(33\!\cdots\!33\)\( T^{8} + \)\(14\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(11\!\cdots\!52\)\( T^{11} - \)\(49\!\cdots\!44\)\( T^{12} - \)\(26\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!76\)\( T^{14} - \)\(26\!\cdots\!40\)\( p^{4} T^{15} - \)\(49\!\cdots\!44\)\( p^{8} T^{16} - \)\(11\!\cdots\!52\)\( p^{12} T^{17} + \)\(13\!\cdots\!72\)\( p^{16} T^{18} + \)\(14\!\cdots\!60\)\( p^{20} T^{19} + \)\(33\!\cdots\!33\)\( p^{24} T^{20} - \)\(63\!\cdots\!30\)\( p^{28} T^{21} - \)\(22\!\cdots\!34\)\( p^{32} T^{22} - 5986165856758813458 p^{36} T^{23} - 394142745410454 p^{40} T^{24} + 203665889578 p^{44} T^{25} + 70876418 p^{48} T^{26} + 11906 p^{52} T^{27} + p^{56} T^{28} \)
67 \( 1 - 184400600 T^{2} + 16859785332405158 T^{4} - \)\(10\!\cdots\!92\)\( T^{6} + \)\(43\!\cdots\!61\)\( T^{8} - \)\(14\!\cdots\!98\)\( T^{10} + \)\(40\!\cdots\!32\)\( T^{12} - \)\(89\!\cdots\!60\)\( T^{14} + \)\(40\!\cdots\!32\)\( p^{8} T^{16} - \)\(14\!\cdots\!98\)\( p^{16} T^{18} + \)\(43\!\cdots\!61\)\( p^{24} T^{20} - \)\(10\!\cdots\!92\)\( p^{32} T^{22} + 16859785332405158 p^{40} T^{24} - 184400600 p^{48} T^{26} + p^{56} T^{28} \)
71 \( ( 1 + 5612 T + 60900748 T^{2} + 483650204234 T^{3} + 2976850829770066 T^{4} + 19760900351284778516 T^{5} + \)\(11\!\cdots\!69\)\( T^{6} + \)\(53\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!69\)\( p^{4} T^{8} + 19760900351284778516 p^{8} T^{9} + 2976850829770066 p^{12} T^{10} + 483650204234 p^{16} T^{11} + 60900748 p^{20} T^{12} + 5612 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
73 \( 1 - 95040550 T^{2} + 4729797140050243 T^{4} - \)\(18\!\cdots\!34\)\( T^{6} + \)\(70\!\cdots\!77\)\( T^{8} - \)\(24\!\cdots\!26\)\( T^{10} + \)\(77\!\cdots\!16\)\( T^{12} - \)\(23\!\cdots\!18\)\( T^{14} + \)\(77\!\cdots\!16\)\( p^{8} T^{16} - \)\(24\!\cdots\!26\)\( p^{16} T^{18} + \)\(70\!\cdots\!77\)\( p^{24} T^{20} - \)\(18\!\cdots\!34\)\( p^{32} T^{22} + 4729797140050243 p^{40} T^{24} - 95040550 p^{48} T^{26} + p^{56} T^{28} \)
79 \( 1 - 20488 T + 209879072 T^{2} - 1422808826512 T^{3} + 7013576890902382 T^{4} - 35007990661500345608 T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!69\)\( T^{8} - \)\(70\!\cdots\!44\)\( p T^{9} + \)\(29\!\cdots\!96\)\( T^{10} - \)\(16\!\cdots\!64\)\( T^{11} + \)\(47\!\cdots\!04\)\( T^{12} + \)\(28\!\cdots\!12\)\( T^{13} - \)\(35\!\cdots\!00\)\( T^{14} + \)\(28\!\cdots\!12\)\( p^{4} T^{15} + \)\(47\!\cdots\!04\)\( p^{8} T^{16} - \)\(16\!\cdots\!64\)\( p^{12} T^{17} + \)\(29\!\cdots\!96\)\( p^{16} T^{18} - \)\(70\!\cdots\!44\)\( p^{21} T^{19} + \)\(11\!\cdots\!69\)\( p^{24} T^{20} - \)\(19\!\cdots\!04\)\( p^{28} T^{21} + \)\(25\!\cdots\!72\)\( p^{32} T^{22} - 35007990661500345608 p^{36} T^{23} + 7013576890902382 p^{40} T^{24} - 1422808826512 p^{44} T^{25} + 209879072 p^{48} T^{26} - 20488 p^{52} T^{27} + p^{56} T^{28} \)
83 \( ( 1 + 10112 T + 225278652 T^{2} + 1598383207850 T^{3} + 22827707456046842 T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!69\)\( T^{6} + \)\(80\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!69\)\( p^{4} T^{8} + \)\(13\!\cdots\!20\)\( p^{8} T^{9} + 22827707456046842 p^{12} T^{10} + 1598383207850 p^{16} T^{11} + 225278652 p^{20} T^{12} + 10112 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
89 \( 1 - 13864 T + 96105248 T^{2} - 1285975306536 T^{3} + 12718246049793051 T^{4} - 31352269526314024464 T^{5} + \)\(39\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{7} - \)\(42\!\cdots\!59\)\( T^{8} + \)\(26\!\cdots\!36\)\( T^{9} - \)\(10\!\cdots\!64\)\( T^{10} + \)\(52\!\cdots\!16\)\( T^{11} + \)\(20\!\cdots\!27\)\( T^{12} - \)\(26\!\cdots\!28\)\( p T^{13} + \)\(17\!\cdots\!88\)\( p^{2} T^{14} - \)\(26\!\cdots\!28\)\( p^{5} T^{15} + \)\(20\!\cdots\!27\)\( p^{8} T^{16} + \)\(52\!\cdots\!16\)\( p^{12} T^{17} - \)\(10\!\cdots\!64\)\( p^{16} T^{18} + \)\(26\!\cdots\!36\)\( p^{20} T^{19} - \)\(42\!\cdots\!59\)\( p^{24} T^{20} + \)\(11\!\cdots\!04\)\( p^{28} T^{21} + \)\(39\!\cdots\!96\)\( p^{32} T^{22} - 31352269526314024464 p^{36} T^{23} + 12718246049793051 p^{40} T^{24} - 1285975306536 p^{44} T^{25} + 96105248 p^{48} T^{26} - 13864 p^{52} T^{27} + p^{56} T^{28} \)
97 \( 1 - 16622 T + 138145442 T^{2} - 1179500170998 T^{3} + 25150805825710411 T^{4} - \)\(37\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!60\)\( T^{6} - \)\(34\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!25\)\( T^{8} - \)\(44\!\cdots\!34\)\( T^{9} + \)\(42\!\cdots\!06\)\( T^{10} - \)\(47\!\cdots\!70\)\( T^{11} + \)\(53\!\cdots\!63\)\( T^{12} - \)\(44\!\cdots\!00\)\( T^{13} + \)\(37\!\cdots\!16\)\( T^{14} - \)\(44\!\cdots\!00\)\( p^{4} T^{15} + \)\(53\!\cdots\!63\)\( p^{8} T^{16} - \)\(47\!\cdots\!70\)\( p^{12} T^{17} + \)\(42\!\cdots\!06\)\( p^{16} T^{18} - \)\(44\!\cdots\!34\)\( p^{20} T^{19} + \)\(40\!\cdots\!25\)\( p^{24} T^{20} - \)\(34\!\cdots\!16\)\( p^{28} T^{21} + \)\(34\!\cdots\!60\)\( p^{32} T^{22} - \)\(37\!\cdots\!60\)\( p^{36} T^{23} + 25150805825710411 p^{40} T^{24} - 1179500170998 p^{44} T^{25} + 138145442 p^{48} T^{26} - 16622 p^{52} T^{27} + p^{56} T^{28} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.88202723334510770710677957698, −3.84942072193650612333145556537, −3.77284861472737160353995953786, −3.62138247176059017561337135656, −3.32084500459170972177967582874, −3.28233409351122936408875641431, −3.16525530725461329986226375207, −2.93936963553850864238812329000, −2.82541742880188423561467186368, −2.45777318112433199143028461860, −2.17613300678973335575849843622, −2.06099650173271153187448753641, −2.01903410027451328849633560125, −1.86728897107088372558998871780, −1.79460100112356694124601538047, −1.73178414000045282133590468960, −1.38520994873677481530998716518, −1.29377642749412454911514237772, −1.28547910699209713373851189589, −1.14663205384707948046140648180, −1.03119008537910743556548709971, −0.76578567989378638861092152207, −0.15493793344326275676906982287, −0.11703406399883178732861041224, −0.00075570786083443713575903326, 0.00075570786083443713575903326, 0.11703406399883178732861041224, 0.15493793344326275676906982287, 0.76578567989378638861092152207, 1.03119008537910743556548709971, 1.14663205384707948046140648180, 1.28547910699209713373851189589, 1.29377642749412454911514237772, 1.38520994873677481530998716518, 1.73178414000045282133590468960, 1.79460100112356694124601538047, 1.86728897107088372558998871780, 2.01903410027451328849633560125, 2.06099650173271153187448753641, 2.17613300678973335575849843622, 2.45777318112433199143028461860, 2.82541742880188423561467186368, 2.93936963553850864238812329000, 3.16525530725461329986226375207, 3.28233409351122936408875641431, 3.32084500459170972177967582874, 3.62138247176059017561337135656, 3.77284861472737160353995953786, 3.84942072193650612333145556537, 3.88202723334510770710677957698

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

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