Properties

Label 28-84e14-1.1-c11e14-0-0
Degree $28$
Conductor $8.708\times 10^{26}$
Sign $1$
Analytic cond. $2.17612\times 10^{25}$
Root an. cond. $8.03373$
Motivic weight $11$
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  + 1.70e3·3-s + 7.21e3·5-s + 3.50e4·7-s + 1.24e6·9-s + 5.44e4·11-s + 1.53e6·13-s + 1.22e7·15-s + 1.47e6·17-s − 2.28e7·19-s + 5.95e7·21-s + 6.25e7·23-s + 1.65e8·25-s + 4.01e8·27-s + 1.02e8·29-s + 1.88e8·31-s + 9.26e7·33-s + 2.52e8·35-s + 1.99e8·37-s + 2.61e9·39-s − 6.93e8·41-s − 6.20e8·43-s + 8.95e9·45-s + 2.77e9·47-s − 1.99e9·49-s + 2.51e9·51-s + 6.48e9·53-s + 3.93e8·55-s + ⋯
L(s)  = 1  + 4.04·3-s + 1.03·5-s + 0.787·7-s + 7·9-s + 0.101·11-s + 1.14·13-s + 4.17·15-s + 0.252·17-s − 2.11·19-s + 3.18·21-s + 2.02·23-s + 3.39·25-s + 5.38·27-s + 0.924·29-s + 1.18·31-s + 0.411·33-s + 0.813·35-s + 0.473·37-s + 4.63·39-s − 0.935·41-s − 0.643·43-s + 7.23·45-s + 1.76·47-s − 1.00·49-s + 1.02·51-s + 2.13·53-s + 0.105·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{28} \cdot 3^{14} \cdot 7^{14}\)
Sign: $1$
Analytic conductor: \(2.17612\times 10^{25}\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{28} \cdot 3^{14} \cdot 7^{14} ,\ ( \ : [11/2]^{14} ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(7.024758942\)
\(L(\frac12)\) \(\approx\) \(7.024758942\)
\(L(\frac{13}{2})\) 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 \( ( 1 - p^{5} T + p^{10} T^{2} )^{7} \)
7 \( 1 - 35001 T + 459586934 p T^{2} - 416640933893 p^{3} T^{3} - 2652179369349 p^{6} T^{4} + 641582460316730 p^{10} T^{5} - 20882864930402416 p^{14} T^{6} + 111890596406780032 p^{19} T^{7} - 20882864930402416 p^{25} T^{8} + 641582460316730 p^{32} T^{9} - 2652179369349 p^{39} T^{10} - 416640933893 p^{47} T^{11} + 459586934 p^{56} T^{12} - 35001 p^{66} T^{13} + p^{77} T^{14} \)
good5 \( 1 - 7218 T - 22756121 p T^{2} + 649156696038 T^{3} + 4307586313318601 T^{4} + 249101754660946656 p T^{5} - \)\(21\!\cdots\!76\)\( p^{3} T^{6} - \)\(49\!\cdots\!28\)\( p^{4} T^{7} + \)\(76\!\cdots\!64\)\( p^{5} T^{8} - \)\(34\!\cdots\!32\)\( p^{5} T^{9} - \)\(12\!\cdots\!04\)\( p^{7} T^{10} + \)\(42\!\cdots\!52\)\( p^{7} T^{11} + \)\(13\!\cdots\!86\)\( p^{8} T^{12} + \)\(77\!\cdots\!48\)\( p^{9} T^{13} - \)\(38\!\cdots\!14\)\( p^{10} T^{14} + \)\(77\!\cdots\!48\)\( p^{20} T^{15} + \)\(13\!\cdots\!86\)\( p^{30} T^{16} + \)\(42\!\cdots\!52\)\( p^{40} T^{17} - \)\(12\!\cdots\!04\)\( p^{51} T^{18} - \)\(34\!\cdots\!32\)\( p^{60} T^{19} + \)\(76\!\cdots\!64\)\( p^{71} T^{20} - \)\(49\!\cdots\!28\)\( p^{81} T^{21} - \)\(21\!\cdots\!76\)\( p^{91} T^{22} + 249101754660946656 p^{100} T^{23} + 4307586313318601 p^{110} T^{24} + 649156696038 p^{121} T^{25} - 22756121 p^{133} T^{26} - 7218 p^{143} T^{27} + p^{154} T^{28} \)
11 \( 1 - 450 p^{2} T - 751585286491 T^{2} - 197242697593561734 T^{3} + \)\(31\!\cdots\!25\)\( T^{4} + \)\(19\!\cdots\!96\)\( T^{5} - \)\(69\!\cdots\!48\)\( p T^{6} - \)\(10\!\cdots\!08\)\( T^{7} - \)\(21\!\cdots\!76\)\( T^{8} + \)\(36\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!36\)\( p T^{10} - \)\(91\!\cdots\!88\)\( T^{11} - \)\(64\!\cdots\!78\)\( T^{12} + \)\(10\!\cdots\!92\)\( T^{13} + \)\(22\!\cdots\!30\)\( T^{14} + \)\(10\!\cdots\!92\)\( p^{11} T^{15} - \)\(64\!\cdots\!78\)\( p^{22} T^{16} - \)\(91\!\cdots\!88\)\( p^{33} T^{17} + \)\(10\!\cdots\!36\)\( p^{45} T^{18} + \)\(36\!\cdots\!96\)\( p^{55} T^{19} - \)\(21\!\cdots\!76\)\( p^{66} T^{20} - \)\(10\!\cdots\!08\)\( p^{77} T^{21} - \)\(69\!\cdots\!48\)\( p^{89} T^{22} + \)\(19\!\cdots\!96\)\( p^{99} T^{23} + \)\(31\!\cdots\!25\)\( p^{110} T^{24} - 197242697593561734 p^{121} T^{25} - 751585286491 p^{132} T^{26} - 450 p^{145} T^{27} + p^{154} T^{28} \)
13 \( ( 1 - 767491 T + 439746164270 p T^{2} - 3692688153433399307 T^{3} + \)\(19\!\cdots\!60\)\( T^{4} - \)\(11\!\cdots\!57\)\( T^{5} + \)\(47\!\cdots\!81\)\( T^{6} - \)\(23\!\cdots\!98\)\( T^{7} + \)\(47\!\cdots\!81\)\( p^{11} T^{8} - \)\(11\!\cdots\!57\)\( p^{22} T^{9} + \)\(19\!\cdots\!60\)\( p^{33} T^{10} - 3692688153433399307 p^{44} T^{11} + 439746164270 p^{56} T^{12} - 767491 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
17 \( 1 - 1478880 T - 86594506886215 T^{2} + \)\(48\!\cdots\!88\)\( T^{3} + \)\(39\!\cdots\!88\)\( T^{4} - \)\(40\!\cdots\!64\)\( T^{5} - \)\(62\!\cdots\!65\)\( T^{6} + \)\(11\!\cdots\!04\)\( p T^{7} - \)\(36\!\cdots\!03\)\( T^{8} - \)\(62\!\cdots\!20\)\( T^{9} + \)\(30\!\cdots\!82\)\( T^{10} + \)\(12\!\cdots\!60\)\( T^{11} - \)\(13\!\cdots\!91\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(45\!\cdots\!23\)\( T^{14} - \)\(14\!\cdots\!08\)\( p^{11} T^{15} - \)\(13\!\cdots\!91\)\( p^{22} T^{16} + \)\(12\!\cdots\!60\)\( p^{33} T^{17} + \)\(30\!\cdots\!82\)\( p^{44} T^{18} - \)\(62\!\cdots\!20\)\( p^{55} T^{19} - \)\(36\!\cdots\!03\)\( p^{66} T^{20} + \)\(11\!\cdots\!04\)\( p^{78} T^{21} - \)\(62\!\cdots\!65\)\( p^{88} T^{22} - \)\(40\!\cdots\!64\)\( p^{99} T^{23} + \)\(39\!\cdots\!88\)\( p^{110} T^{24} + \)\(48\!\cdots\!88\)\( p^{121} T^{25} - 86594506886215 p^{132} T^{26} - 1478880 p^{143} T^{27} + p^{154} T^{28} \)
19 \( 1 + 22875935 T - 126366098263455 T^{2} - \)\(59\!\cdots\!90\)\( T^{3} + \)\(32\!\cdots\!57\)\( T^{4} + \)\(78\!\cdots\!45\)\( T^{5} + \)\(19\!\cdots\!52\)\( T^{6} - \)\(27\!\cdots\!77\)\( p T^{7} + \)\(25\!\cdots\!60\)\( T^{8} + \)\(24\!\cdots\!65\)\( T^{9} - \)\(33\!\cdots\!40\)\( T^{10} - \)\(32\!\cdots\!47\)\( T^{11} + \)\(19\!\cdots\!54\)\( T^{12} + \)\(11\!\cdots\!53\)\( p T^{13} + \)\(15\!\cdots\!98\)\( p^{2} T^{14} + \)\(11\!\cdots\!53\)\( p^{12} T^{15} + \)\(19\!\cdots\!54\)\( p^{22} T^{16} - \)\(32\!\cdots\!47\)\( p^{33} T^{17} - \)\(33\!\cdots\!40\)\( p^{44} T^{18} + \)\(24\!\cdots\!65\)\( p^{55} T^{19} + \)\(25\!\cdots\!60\)\( p^{66} T^{20} - \)\(27\!\cdots\!77\)\( p^{78} T^{21} + \)\(19\!\cdots\!52\)\( p^{88} T^{22} + \)\(78\!\cdots\!45\)\( p^{99} T^{23} + \)\(32\!\cdots\!57\)\( p^{110} T^{24} - \)\(59\!\cdots\!90\)\( p^{121} T^{25} - 126366098263455 p^{132} T^{26} + 22875935 p^{143} T^{27} + p^{154} T^{28} \)
23 \( 1 - 62540568 T - 2320547607442369 T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(27\!\cdots\!96\)\( T^{4} - \)\(41\!\cdots\!84\)\( T^{5} - \)\(20\!\cdots\!47\)\( T^{6} + \)\(54\!\cdots\!68\)\( T^{7} + \)\(15\!\cdots\!09\)\( T^{8} - \)\(52\!\cdots\!40\)\( T^{9} - \)\(26\!\cdots\!98\)\( T^{10} + \)\(36\!\cdots\!16\)\( T^{11} + \)\(47\!\cdots\!77\)\( T^{12} - \)\(12\!\cdots\!96\)\( T^{13} - \)\(55\!\cdots\!03\)\( T^{14} - \)\(12\!\cdots\!96\)\( p^{11} T^{15} + \)\(47\!\cdots\!77\)\( p^{22} T^{16} + \)\(36\!\cdots\!16\)\( p^{33} T^{17} - \)\(26\!\cdots\!98\)\( p^{44} T^{18} - \)\(52\!\cdots\!40\)\( p^{55} T^{19} + \)\(15\!\cdots\!09\)\( p^{66} T^{20} + \)\(54\!\cdots\!68\)\( p^{77} T^{21} - \)\(20\!\cdots\!47\)\( p^{88} T^{22} - \)\(41\!\cdots\!84\)\( p^{99} T^{23} + \)\(27\!\cdots\!96\)\( p^{110} T^{24} + \)\(21\!\cdots\!64\)\( p^{121} T^{25} - 2320547607442369 p^{132} T^{26} - 62540568 p^{143} T^{27} + p^{154} T^{28} \)
29 \( ( 1 - 51048864 T + 60092931461407477 T^{2} - \)\(18\!\cdots\!48\)\( T^{3} + \)\(17\!\cdots\!76\)\( T^{4} - \)\(37\!\cdots\!52\)\( T^{5} + \)\(30\!\cdots\!46\)\( T^{6} - \)\(52\!\cdots\!52\)\( T^{7} + \)\(30\!\cdots\!46\)\( p^{11} T^{8} - \)\(37\!\cdots\!52\)\( p^{22} T^{9} + \)\(17\!\cdots\!76\)\( p^{33} T^{10} - \)\(18\!\cdots\!48\)\( p^{44} T^{11} + 60092931461407477 p^{55} T^{12} - 51048864 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
31 \( 1 - 188600405 T - 63403647727147201 T^{2} + \)\(26\!\cdots\!20\)\( T^{3} - \)\(16\!\cdots\!54\)\( T^{4} - \)\(11\!\cdots\!50\)\( T^{5} + \)\(18\!\cdots\!11\)\( T^{6} + \)\(14\!\cdots\!95\)\( T^{7} - \)\(76\!\cdots\!56\)\( T^{8} + \)\(86\!\cdots\!95\)\( T^{9} + \)\(83\!\cdots\!01\)\( T^{10} - \)\(39\!\cdots\!50\)\( T^{11} + \)\(42\!\cdots\!21\)\( T^{12} + \)\(52\!\cdots\!45\)\( T^{13} - \)\(18\!\cdots\!46\)\( T^{14} + \)\(52\!\cdots\!45\)\( p^{11} T^{15} + \)\(42\!\cdots\!21\)\( p^{22} T^{16} - \)\(39\!\cdots\!50\)\( p^{33} T^{17} + \)\(83\!\cdots\!01\)\( p^{44} T^{18} + \)\(86\!\cdots\!95\)\( p^{55} T^{19} - \)\(76\!\cdots\!56\)\( p^{66} T^{20} + \)\(14\!\cdots\!95\)\( p^{77} T^{21} + \)\(18\!\cdots\!11\)\( p^{88} T^{22} - \)\(11\!\cdots\!50\)\( p^{99} T^{23} - \)\(16\!\cdots\!54\)\( p^{110} T^{24} + \)\(26\!\cdots\!20\)\( p^{121} T^{25} - 63403647727147201 p^{132} T^{26} - 188600405 p^{143} T^{27} + p^{154} T^{28} \)
37 \( 1 - 199685599 T - 424081657589527005 T^{2} + \)\(12\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(18\!\cdots\!63\)\( T^{5} + \)\(56\!\cdots\!76\)\( T^{6} - \)\(50\!\cdots\!57\)\( T^{7} + \)\(22\!\cdots\!30\)\( T^{8} - \)\(17\!\cdots\!11\)\( T^{9} - \)\(86\!\cdots\!32\)\( T^{10} + \)\(73\!\cdots\!23\)\( T^{11} + \)\(69\!\cdots\!32\)\( T^{12} - \)\(43\!\cdots\!05\)\( T^{13} + \)\(82\!\cdots\!70\)\( T^{14} - \)\(43\!\cdots\!05\)\( p^{11} T^{15} + \)\(69\!\cdots\!32\)\( p^{22} T^{16} + \)\(73\!\cdots\!23\)\( p^{33} T^{17} - \)\(86\!\cdots\!32\)\( p^{44} T^{18} - \)\(17\!\cdots\!11\)\( p^{55} T^{19} + \)\(22\!\cdots\!30\)\( p^{66} T^{20} - \)\(50\!\cdots\!57\)\( p^{77} T^{21} + \)\(56\!\cdots\!76\)\( p^{88} T^{22} - \)\(18\!\cdots\!63\)\( p^{99} T^{23} + \)\(24\!\cdots\!95\)\( p^{110} T^{24} + \)\(12\!\cdots\!56\)\( p^{121} T^{25} - 424081657589527005 p^{132} T^{26} - 199685599 p^{143} T^{27} + p^{154} T^{28} \)
41 \( ( 1 + 346934358 T + 2666136544004744267 T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!01\)\( T^{4} + \)\(13\!\cdots\!86\)\( T^{5} + \)\(25\!\cdots\!75\)\( T^{6} + \)\(94\!\cdots\!44\)\( T^{7} + \)\(25\!\cdots\!75\)\( p^{11} T^{8} + \)\(13\!\cdots\!86\)\( p^{22} T^{9} + \)\(33\!\cdots\!01\)\( p^{33} T^{10} + \)\(10\!\cdots\!40\)\( p^{44} T^{11} + 2666136544004744267 p^{55} T^{12} + 346934358 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
43 \( ( 1 + 310350877 T + 4110189730212486712 T^{2} + \)\(27\!\cdots\!05\)\( T^{3} + \)\(74\!\cdots\!30\)\( T^{4} + \)\(75\!\cdots\!03\)\( T^{5} + \)\(85\!\cdots\!87\)\( T^{6} + \)\(97\!\cdots\!34\)\( T^{7} + \)\(85\!\cdots\!87\)\( p^{11} T^{8} + \)\(75\!\cdots\!03\)\( p^{22} T^{9} + \)\(74\!\cdots\!30\)\( p^{33} T^{10} + \)\(27\!\cdots\!05\)\( p^{44} T^{11} + 4110189730212486712 p^{55} T^{12} + 310350877 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
47 \( 1 - 2771987346 T - 3981206422535827625 T^{2} + \)\(21\!\cdots\!70\)\( T^{3} - \)\(65\!\cdots\!56\)\( T^{4} - \)\(70\!\cdots\!46\)\( T^{5} + \)\(71\!\cdots\!93\)\( T^{6} + \)\(14\!\cdots\!02\)\( T^{7} - \)\(25\!\cdots\!83\)\( T^{8} - \)\(22\!\cdots\!36\)\( T^{9} + \)\(68\!\cdots\!70\)\( T^{10} + \)\(21\!\cdots\!92\)\( T^{11} - \)\(14\!\cdots\!07\)\( T^{12} - \)\(51\!\cdots\!86\)\( T^{13} + \)\(32\!\cdots\!93\)\( T^{14} - \)\(51\!\cdots\!86\)\( p^{11} T^{15} - \)\(14\!\cdots\!07\)\( p^{22} T^{16} + \)\(21\!\cdots\!92\)\( p^{33} T^{17} + \)\(68\!\cdots\!70\)\( p^{44} T^{18} - \)\(22\!\cdots\!36\)\( p^{55} T^{19} - \)\(25\!\cdots\!83\)\( p^{66} T^{20} + \)\(14\!\cdots\!02\)\( p^{77} T^{21} + \)\(71\!\cdots\!93\)\( p^{88} T^{22} - \)\(70\!\cdots\!46\)\( p^{99} T^{23} - \)\(65\!\cdots\!56\)\( p^{110} T^{24} + \)\(21\!\cdots\!70\)\( p^{121} T^{25} - 3981206422535827625 p^{132} T^{26} - 2771987346 p^{143} T^{27} + p^{154} T^{28} \)
53 \( 1 - 6487034184 T - 17804802935623445237 T^{2} + \)\(19\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!93\)\( T^{4} - \)\(39\!\cdots\!04\)\( T^{5} + \)\(82\!\cdots\!88\)\( T^{6} + \)\(54\!\cdots\!16\)\( T^{7} - \)\(66\!\cdots\!60\)\( T^{8} - \)\(56\!\cdots\!12\)\( T^{9} + \)\(13\!\cdots\!88\)\( T^{10} + \)\(42\!\cdots\!92\)\( T^{11} - \)\(19\!\cdots\!22\)\( T^{12} - \)\(14\!\cdots\!16\)\( T^{13} + \)\(20\!\cdots\!06\)\( T^{14} - \)\(14\!\cdots\!16\)\( p^{11} T^{15} - \)\(19\!\cdots\!22\)\( p^{22} T^{16} + \)\(42\!\cdots\!92\)\( p^{33} T^{17} + \)\(13\!\cdots\!88\)\( p^{44} T^{18} - \)\(56\!\cdots\!12\)\( p^{55} T^{19} - \)\(66\!\cdots\!60\)\( p^{66} T^{20} + \)\(54\!\cdots\!16\)\( p^{77} T^{21} + \)\(82\!\cdots\!88\)\( p^{88} T^{22} - \)\(39\!\cdots\!04\)\( p^{99} T^{23} + \)\(18\!\cdots\!93\)\( p^{110} T^{24} + \)\(19\!\cdots\!56\)\( p^{121} T^{25} - 17804802935623445237 p^{132} T^{26} - 6487034184 p^{143} T^{27} + p^{154} T^{28} \)
59 \( 1 + 8183838888 T - 81786955029453871555 T^{2} - \)\(10\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!93\)\( T^{4} + \)\(52\!\cdots\!24\)\( T^{5} + \)\(45\!\cdots\!20\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} - \)\(14\!\cdots\!36\)\( T^{8} + \)\(41\!\cdots\!76\)\( T^{9} - \)\(78\!\cdots\!56\)\( T^{10} - \)\(15\!\cdots\!52\)\( T^{11} - \)\(20\!\cdots\!66\)\( T^{12} + \)\(24\!\cdots\!44\)\( T^{13} + \)\(98\!\cdots\!74\)\( T^{14} + \)\(24\!\cdots\!44\)\( p^{11} T^{15} - \)\(20\!\cdots\!66\)\( p^{22} T^{16} - \)\(15\!\cdots\!52\)\( p^{33} T^{17} - \)\(78\!\cdots\!56\)\( p^{44} T^{18} + \)\(41\!\cdots\!76\)\( p^{55} T^{19} - \)\(14\!\cdots\!36\)\( p^{66} T^{20} - \)\(14\!\cdots\!80\)\( p^{77} T^{21} + \)\(45\!\cdots\!20\)\( p^{88} T^{22} + \)\(52\!\cdots\!24\)\( p^{99} T^{23} + \)\(13\!\cdots\!93\)\( p^{110} T^{24} - \)\(10\!\cdots\!84\)\( p^{121} T^{25} - 81786955029453871555 p^{132} T^{26} + 8183838888 p^{143} T^{27} + p^{154} T^{28} \)
61 \( 1 - 4069556330 T - \)\(10\!\cdots\!99\)\( T^{2} - \)\(45\!\cdots\!62\)\( T^{3} + \)\(62\!\cdots\!84\)\( T^{4} + \)\(68\!\cdots\!58\)\( T^{5} - \)\(50\!\cdots\!89\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{7} - \)\(13\!\cdots\!95\)\( T^{8} + \)\(70\!\cdots\!52\)\( T^{9} + \)\(66\!\cdots\!18\)\( T^{10} - \)\(66\!\cdots\!88\)\( T^{11} - \)\(18\!\cdots\!59\)\( T^{12} + \)\(27\!\cdots\!70\)\( T^{13} + \)\(72\!\cdots\!23\)\( T^{14} + \)\(27\!\cdots\!70\)\( p^{11} T^{15} - \)\(18\!\cdots\!59\)\( p^{22} T^{16} - \)\(66\!\cdots\!88\)\( p^{33} T^{17} + \)\(66\!\cdots\!18\)\( p^{44} T^{18} + \)\(70\!\cdots\!52\)\( p^{55} T^{19} - \)\(13\!\cdots\!95\)\( p^{66} T^{20} - \)\(32\!\cdots\!30\)\( p^{77} T^{21} - \)\(50\!\cdots\!89\)\( p^{88} T^{22} + \)\(68\!\cdots\!58\)\( p^{99} T^{23} + \)\(62\!\cdots\!84\)\( p^{110} T^{24} - \)\(45\!\cdots\!62\)\( p^{121} T^{25} - \)\(10\!\cdots\!99\)\( p^{132} T^{26} - 4069556330 p^{143} T^{27} + p^{154} T^{28} \)
67 \( 1 - 15766443531 T - \)\(42\!\cdots\!91\)\( T^{2} + \)\(87\!\cdots\!18\)\( T^{3} + \)\(85\!\cdots\!17\)\( T^{4} - \)\(23\!\cdots\!69\)\( T^{5} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(37\!\cdots\!15\)\( T^{7} + \)\(14\!\cdots\!00\)\( T^{8} - \)\(42\!\cdots\!85\)\( T^{9} - \)\(29\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!79\)\( T^{11} + \)\(57\!\cdots\!22\)\( T^{12} - \)\(12\!\cdots\!79\)\( T^{13} - \)\(82\!\cdots\!70\)\( T^{14} - \)\(12\!\cdots\!79\)\( p^{11} T^{15} + \)\(57\!\cdots\!22\)\( p^{22} T^{16} + \)\(32\!\cdots\!79\)\( p^{33} T^{17} - \)\(29\!\cdots\!12\)\( p^{44} T^{18} - \)\(42\!\cdots\!85\)\( p^{55} T^{19} + \)\(14\!\cdots\!00\)\( p^{66} T^{20} + \)\(37\!\cdots\!15\)\( p^{77} T^{21} - \)\(10\!\cdots\!64\)\( p^{88} T^{22} - \)\(23\!\cdots\!69\)\( p^{99} T^{23} + \)\(85\!\cdots\!17\)\( p^{110} T^{24} + \)\(87\!\cdots\!18\)\( p^{121} T^{25} - \)\(42\!\cdots\!91\)\( p^{132} T^{26} - 15766443531 p^{143} T^{27} + p^{154} T^{28} \)
71 \( ( 1 + 16591642722 T + \)\(12\!\cdots\!93\)\( T^{2} + \)\(20\!\cdots\!16\)\( T^{3} + \)\(77\!\cdots\!25\)\( T^{4} + \)\(10\!\cdots\!30\)\( T^{5} + \)\(27\!\cdots\!17\)\( T^{6} + \)\(32\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!17\)\( p^{11} T^{8} + \)\(10\!\cdots\!30\)\( p^{22} T^{9} + \)\(77\!\cdots\!25\)\( p^{33} T^{10} + \)\(20\!\cdots\!16\)\( p^{44} T^{11} + \)\(12\!\cdots\!93\)\( p^{55} T^{12} + 16591642722 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
73 \( 1 + 31685143839 T - \)\(61\!\cdots\!33\)\( T^{2} - \)\(25\!\cdots\!24\)\( T^{3} + \)\(21\!\cdots\!19\)\( T^{4} + \)\(10\!\cdots\!03\)\( T^{5} - \)\(87\!\cdots\!80\)\( T^{6} - \)\(59\!\cdots\!83\)\( p T^{7} + \)\(13\!\cdots\!62\)\( T^{8} + \)\(13\!\cdots\!67\)\( T^{9} + \)\(12\!\cdots\!44\)\( T^{10} - \)\(33\!\cdots\!51\)\( T^{11} - \)\(82\!\cdots\!52\)\( T^{12} + \)\(46\!\cdots\!33\)\( T^{13} + \)\(43\!\cdots\!50\)\( T^{14} + \)\(46\!\cdots\!33\)\( p^{11} T^{15} - \)\(82\!\cdots\!52\)\( p^{22} T^{16} - \)\(33\!\cdots\!51\)\( p^{33} T^{17} + \)\(12\!\cdots\!44\)\( p^{44} T^{18} + \)\(13\!\cdots\!67\)\( p^{55} T^{19} + \)\(13\!\cdots\!62\)\( p^{66} T^{20} - \)\(59\!\cdots\!83\)\( p^{78} T^{21} - \)\(87\!\cdots\!80\)\( p^{88} T^{22} + \)\(10\!\cdots\!03\)\( p^{99} T^{23} + \)\(21\!\cdots\!19\)\( p^{110} T^{24} - \)\(25\!\cdots\!24\)\( p^{121} T^{25} - \)\(61\!\cdots\!33\)\( p^{132} T^{26} + 31685143839 p^{143} T^{27} + p^{154} T^{28} \)
79 \( 1 - 21999509987 T - \)\(26\!\cdots\!93\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!90\)\( T^{4} - \)\(25\!\cdots\!14\)\( T^{5} - \)\(42\!\cdots\!45\)\( T^{6} + \)\(34\!\cdots\!45\)\( T^{7} - \)\(42\!\cdots\!92\)\( T^{8} - \)\(32\!\cdots\!75\)\( T^{9} + \)\(84\!\cdots\!85\)\( T^{10} + \)\(20\!\cdots\!34\)\( T^{11} - \)\(10\!\cdots\!59\)\( T^{12} - \)\(62\!\cdots\!17\)\( T^{13} + \)\(86\!\cdots\!74\)\( T^{14} - \)\(62\!\cdots\!17\)\( p^{11} T^{15} - \)\(10\!\cdots\!59\)\( p^{22} T^{16} + \)\(20\!\cdots\!34\)\( p^{33} T^{17} + \)\(84\!\cdots\!85\)\( p^{44} T^{18} - \)\(32\!\cdots\!75\)\( p^{55} T^{19} - \)\(42\!\cdots\!92\)\( p^{66} T^{20} + \)\(34\!\cdots\!45\)\( p^{77} T^{21} - \)\(42\!\cdots\!45\)\( p^{88} T^{22} - \)\(25\!\cdots\!14\)\( p^{99} T^{23} + \)\(30\!\cdots\!90\)\( p^{110} T^{24} + \)\(11\!\cdots\!60\)\( p^{121} T^{25} - \)\(26\!\cdots\!93\)\( p^{132} T^{26} - 21999509987 p^{143} T^{27} + p^{154} T^{28} \)
83 \( ( 1 + 31526942994 T + \)\(33\!\cdots\!15\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!20\)\( T^{4} + \)\(23\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!58\)\( T^{6} + \)\(23\!\cdots\!08\)\( T^{7} + \)\(69\!\cdots\!58\)\( p^{11} T^{8} + \)\(23\!\cdots\!20\)\( p^{22} T^{9} + \)\(50\!\cdots\!20\)\( p^{33} T^{10} + \)\(15\!\cdots\!76\)\( p^{44} T^{11} + \)\(33\!\cdots\!15\)\( p^{55} T^{12} + 31526942994 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
89 \( 1 - 67041904680 T - \)\(73\!\cdots\!07\)\( T^{2} + \)\(37\!\cdots\!04\)\( T^{3} + \)\(29\!\cdots\!68\)\( T^{4} - \)\(42\!\cdots\!60\)\( p T^{5} - \)\(13\!\cdots\!01\)\( T^{6} + \)\(87\!\cdots\!68\)\( T^{7} + \)\(48\!\cdots\!53\)\( T^{8} - \)\(91\!\cdots\!68\)\( T^{9} - \)\(13\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(48\!\cdots\!65\)\( T^{12} + \)\(13\!\cdots\!48\)\( T^{13} - \)\(16\!\cdots\!05\)\( T^{14} + \)\(13\!\cdots\!48\)\( p^{11} T^{15} + \)\(48\!\cdots\!65\)\( p^{22} T^{16} + \)\(10\!\cdots\!80\)\( p^{33} T^{17} - \)\(13\!\cdots\!86\)\( p^{44} T^{18} - \)\(91\!\cdots\!68\)\( p^{55} T^{19} + \)\(48\!\cdots\!53\)\( p^{66} T^{20} + \)\(87\!\cdots\!68\)\( p^{77} T^{21} - \)\(13\!\cdots\!01\)\( p^{88} T^{22} - \)\(42\!\cdots\!60\)\( p^{100} T^{23} + \)\(29\!\cdots\!68\)\( p^{110} T^{24} + \)\(37\!\cdots\!04\)\( p^{121} T^{25} - \)\(73\!\cdots\!07\)\( p^{132} T^{26} - 67041904680 p^{143} T^{27} + p^{154} T^{28} \)
97 \( ( 1 - 142041709050 T + \)\(29\!\cdots\!69\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(36\!\cdots\!28\)\( T^{4} - \)\(27\!\cdots\!80\)\( T^{5} + \)\(38\!\cdots\!86\)\( T^{6} - \)\(25\!\cdots\!40\)\( T^{7} + \)\(38\!\cdots\!86\)\( p^{11} T^{8} - \)\(27\!\cdots\!80\)\( p^{22} T^{9} + \)\(36\!\cdots\!28\)\( p^{33} T^{10} - \)\(24\!\cdots\!80\)\( p^{44} T^{11} + \)\(29\!\cdots\!69\)\( p^{55} T^{12} - 142041709050 p^{66} T^{13} + p^{77} T^{14} )^{2} \)
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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

−2.67043570981259821115952619733, −2.54928247642773050663368440873, −2.49933387124864340018805036810, −2.28468101521223872728359877438, −2.11269781163564467853030056675, −2.07891206508043129527505859256, −2.05124000743498111121789936774, −1.97801203018024465842884105317, −1.86736636016487048997565712840, −1.68688932522544214295482819445, −1.66419572309136207125663773384, −1.60495377198419842295818932675, −1.40541170970683184214453131572, −1.22472210760582836091051954692, −1.19812583088249773038130152614, −1.04085947879610967968273778021, −0.978711925869549678823631075438, −0.941647236502738909937926852604, −0.791603545509958830342488712927, −0.65237817751412660617479176306, −0.61110563088102207314817588397, −0.53147099546546612790719455722, −0.35955066153720175805037634727, −0.07623777328390786046880294659, −0.04304106017020957591427680283, 0.04304106017020957591427680283, 0.07623777328390786046880294659, 0.35955066153720175805037634727, 0.53147099546546612790719455722, 0.61110563088102207314817588397, 0.65237817751412660617479176306, 0.791603545509958830342488712927, 0.941647236502738909937926852604, 0.978711925869549678823631075438, 1.04085947879610967968273778021, 1.19812583088249773038130152614, 1.22472210760582836091051954692, 1.40541170970683184214453131572, 1.60495377198419842295818932675, 1.66419572309136207125663773384, 1.68688932522544214295482819445, 1.86736636016487048997565712840, 1.97801203018024465842884105317, 2.05124000743498111121789936774, 2.07891206508043129527505859256, 2.11269781163564467853030056675, 2.28468101521223872728359877438, 2.49933387124864340018805036810, 2.54928247642773050663368440873, 2.67043570981259821115952619733

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

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