Properties

Label 20-825e10-1.1-c5e10-0-0
Degree $20$
Conductor $1.461\times 10^{29}$
Sign $1$
Analytic cond. $1.64492\times 10^{21}$
Root an. cond. $11.5028$
Motivic weight $5$
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  + 2-s − 90·3-s − 48·4-s − 90·6-s − 188·7-s − 118·8-s + 4.45e3·9-s + 1.21e3·11-s + 4.32e3·12-s + 1.10e3·13-s − 188·14-s + 1.05e3·16-s − 1.10e3·17-s + 4.45e3·18-s + 2.58e3·19-s + 1.69e4·21-s + 1.21e3·22-s − 2.20e3·23-s + 1.06e4·24-s + 1.10e3·26-s − 1.60e5·27-s + 9.02e3·28-s + 5.82e3·29-s − 4.58e3·31-s + 7.74e3·32-s − 1.08e5·33-s − 1.10e3·34-s + ⋯
L(s)  = 1  + 0.176·2-s − 5.77·3-s − 3/2·4-s − 1.02·6-s − 1.45·7-s − 0.651·8-s + 55/3·9-s + 3.01·11-s + 8.66·12-s + 1.80·13-s − 0.256·14-s + 1.02·16-s − 0.928·17-s + 3.24·18-s + 1.64·19-s + 8.37·21-s + 0.533·22-s − 0.869·23-s + 3.76·24-s + 0.319·26-s − 42.3·27-s + 2.17·28-s + 1.28·29-s − 0.857·31-s + 1.33·32-s − 17.4·33-s − 0.164·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{10} \cdot 5^{20} \cdot 11^{10}\)
Sign: $1$
Analytic conductor: \(1.64492\times 10^{21}\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{10} \cdot 5^{20} \cdot 11^{10} ,\ ( \ : [5/2]^{10} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.08905186861\)
\(L(\frac12)\) \(\approx\) \(0.08905186861\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{2} T )^{10} \)
5 \( 1 \)
11 \( ( 1 - p^{2} T )^{10} \)
good2 \( 1 - T + 49 T^{2} + 21 T^{3} + 145 p^{3} T^{4} - 529 p T^{5} + 8849 p^{2} T^{6} - 167 p^{9} T^{7} + 23741 p^{5} T^{8} - 43549 p^{5} T^{9} + 147877 p^{6} T^{10} - 43549 p^{10} T^{11} + 23741 p^{15} T^{12} - 167 p^{24} T^{13} + 8849 p^{22} T^{14} - 529 p^{26} T^{15} + 145 p^{33} T^{16} + 21 p^{35} T^{17} + 49 p^{40} T^{18} - p^{45} T^{19} + p^{50} T^{20} \)
7 \( 1 + 188 T + 67416 T^{2} + 10100696 T^{3} + 2252448197 T^{4} + 272159639540 T^{5} + 53787491320058 T^{6} + 6104708679104920 T^{7} + 164646489361545591 p T^{8} + 2599307046897139724 p^{2} T^{9} + 63182671876300788662 p^{3} T^{10} + 2599307046897139724 p^{7} T^{11} + 164646489361545591 p^{11} T^{12} + 6104708679104920 p^{15} T^{13} + 53787491320058 p^{20} T^{14} + 272159639540 p^{25} T^{15} + 2252448197 p^{30} T^{16} + 10100696 p^{35} T^{17} + 67416 p^{40} T^{18} + 188 p^{45} T^{19} + p^{50} T^{20} \)
13 \( 1 - 1102 T + 1850750 T^{2} - 1494358120 T^{3} + 1522345508624 T^{4} - 961224325025934 T^{5} + 828819561161157786 T^{6} - \)\(46\!\cdots\!70\)\( T^{7} + \)\(38\!\cdots\!83\)\( T^{8} - \)\(20\!\cdots\!82\)\( T^{9} + \)\(15\!\cdots\!40\)\( T^{10} - \)\(20\!\cdots\!82\)\( p^{5} T^{11} + \)\(38\!\cdots\!83\)\( p^{10} T^{12} - \)\(46\!\cdots\!70\)\( p^{15} T^{13} + 828819561161157786 p^{20} T^{14} - 961224325025934 p^{25} T^{15} + 1522345508624 p^{30} T^{16} - 1494358120 p^{35} T^{17} + 1850750 p^{40} T^{18} - 1102 p^{45} T^{19} + p^{50} T^{20} \)
17 \( 1 + 1106 T + 5794111 T^{2} + 5567890534 T^{3} + 18324129473584 T^{4} + 15977992506943590 T^{5} + 41862609393035657193 T^{6} + \)\(32\!\cdots\!26\)\( T^{7} + \)\(76\!\cdots\!03\)\( T^{8} + \)\(53\!\cdots\!24\)\( T^{9} + \)\(11\!\cdots\!92\)\( T^{10} + \)\(53\!\cdots\!24\)\( p^{5} T^{11} + \)\(76\!\cdots\!03\)\( p^{10} T^{12} + \)\(32\!\cdots\!26\)\( p^{15} T^{13} + 41862609393035657193 p^{20} T^{14} + 15977992506943590 p^{25} T^{15} + 18324129473584 p^{30} T^{16} + 5567890534 p^{35} T^{17} + 5794111 p^{40} T^{18} + 1106 p^{45} T^{19} + p^{50} T^{20} \)
19 \( 1 - 2586 T + 17599501 T^{2} - 37072580350 T^{3} + 146669255583219 T^{4} - 259324712037137348 T^{5} + \)\(76\!\cdots\!30\)\( T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!93\)\( T^{8} - \)\(38\!\cdots\!66\)\( T^{9} + \)\(42\!\cdots\!73\)\( p T^{10} - \)\(38\!\cdots\!66\)\( p^{5} T^{11} + \)\(28\!\cdots\!93\)\( p^{10} T^{12} - \)\(11\!\cdots\!40\)\( p^{15} T^{13} + \)\(76\!\cdots\!30\)\( p^{20} T^{14} - 259324712037137348 p^{25} T^{15} + 146669255583219 p^{30} T^{16} - 37072580350 p^{35} T^{17} + 17599501 p^{40} T^{18} - 2586 p^{45} T^{19} + p^{50} T^{20} \)
23 \( 1 + 2206 T + 32365060 T^{2} + 37422934540 T^{3} + 478562660099601 T^{4} + 190856851208522416 T^{5} + \)\(20\!\cdots\!18\)\( p T^{6} - \)\(10\!\cdots\!70\)\( T^{7} + \)\(36\!\cdots\!45\)\( T^{8} - \)\(22\!\cdots\!08\)\( T^{9} + \)\(24\!\cdots\!26\)\( T^{10} - \)\(22\!\cdots\!08\)\( p^{5} T^{11} + \)\(36\!\cdots\!45\)\( p^{10} T^{12} - \)\(10\!\cdots\!70\)\( p^{15} T^{13} + \)\(20\!\cdots\!18\)\( p^{21} T^{14} + 190856851208522416 p^{25} T^{15} + 478562660099601 p^{30} T^{16} + 37422934540 p^{35} T^{17} + 32365060 p^{40} T^{18} + 2206 p^{45} T^{19} + p^{50} T^{20} \)
29 \( 1 - 5824 T + 64168553 T^{2} - 214081424388 T^{3} + 2001929739096353 T^{4} - 8702181550899311980 T^{5} + \)\(71\!\cdots\!48\)\( T^{6} - \)\(10\!\cdots\!88\)\( p T^{7} + \)\(17\!\cdots\!90\)\( T^{8} - \)\(69\!\cdots\!52\)\( T^{9} + \)\(36\!\cdots\!10\)\( T^{10} - \)\(69\!\cdots\!52\)\( p^{5} T^{11} + \)\(17\!\cdots\!90\)\( p^{10} T^{12} - \)\(10\!\cdots\!88\)\( p^{16} T^{13} + \)\(71\!\cdots\!48\)\( p^{20} T^{14} - 8702181550899311980 p^{25} T^{15} + 2001929739096353 p^{30} T^{16} - 214081424388 p^{35} T^{17} + 64168553 p^{40} T^{18} - 5824 p^{45} T^{19} + p^{50} T^{20} \)
31 \( 1 + 4586 T + 132275714 T^{2} + 668005309756 T^{3} + 9057912002512616 T^{4} + 44674313821027703182 T^{5} + \)\(44\!\cdots\!26\)\( T^{6} + \)\(19\!\cdots\!22\)\( T^{7} + \)\(17\!\cdots\!23\)\( T^{8} + \)\(69\!\cdots\!62\)\( T^{9} + \)\(53\!\cdots\!24\)\( T^{10} + \)\(69\!\cdots\!62\)\( p^{5} T^{11} + \)\(17\!\cdots\!23\)\( p^{10} T^{12} + \)\(19\!\cdots\!22\)\( p^{15} T^{13} + \)\(44\!\cdots\!26\)\( p^{20} T^{14} + 44674313821027703182 p^{25} T^{15} + 9057912002512616 p^{30} T^{16} + 668005309756 p^{35} T^{17} + 132275714 p^{40} T^{18} + 4586 p^{45} T^{19} + p^{50} T^{20} \)
37 \( 1 - 18362 T + 487042369 T^{2} - 6928708813522 T^{3} + 110240489601008076 T^{4} - \)\(12\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!27\)\( T^{6} - \)\(15\!\cdots\!78\)\( T^{7} + \)\(16\!\cdots\!51\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!16\)\( T^{10} - \)\(14\!\cdots\!00\)\( p^{5} T^{11} + \)\(16\!\cdots\!51\)\( p^{10} T^{12} - \)\(15\!\cdots\!78\)\( p^{15} T^{13} + \)\(15\!\cdots\!27\)\( p^{20} T^{14} - \)\(12\!\cdots\!14\)\( p^{25} T^{15} + 110240489601008076 p^{30} T^{16} - 6928708813522 p^{35} T^{17} + 487042369 p^{40} T^{18} - 18362 p^{45} T^{19} + p^{50} T^{20} \)
41 \( 1 + 5474 T + 767046899 T^{2} + 3333121404698 T^{3} + 282163687806661324 T^{4} + \)\(91\!\cdots\!34\)\( T^{5} + \)\(66\!\cdots\!81\)\( T^{6} + \)\(15\!\cdots\!30\)\( T^{7} + \)\(11\!\cdots\!11\)\( T^{8} + \)\(20\!\cdots\!64\)\( T^{9} + \)\(14\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!64\)\( p^{5} T^{11} + \)\(11\!\cdots\!11\)\( p^{10} T^{12} + \)\(15\!\cdots\!30\)\( p^{15} T^{13} + \)\(66\!\cdots\!81\)\( p^{20} T^{14} + \)\(91\!\cdots\!34\)\( p^{25} T^{15} + 282163687806661324 p^{30} T^{16} + 3333121404698 p^{35} T^{17} + 767046899 p^{40} T^{18} + 5474 p^{45} T^{19} + p^{50} T^{20} \)
43 \( 1 + 20496 T + 1180446184 T^{2} + 18591100152028 T^{3} + 14635098273903618 p T^{4} + \)\(82\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!66\)\( T^{6} + \)\(23\!\cdots\!12\)\( T^{7} + \)\(48\!\cdots\!01\)\( T^{8} + \)\(46\!\cdots\!56\)\( T^{9} + \)\(82\!\cdots\!72\)\( T^{10} + \)\(46\!\cdots\!56\)\( p^{5} T^{11} + \)\(48\!\cdots\!01\)\( p^{10} T^{12} + \)\(23\!\cdots\!12\)\( p^{15} T^{13} + \)\(20\!\cdots\!66\)\( p^{20} T^{14} + \)\(82\!\cdots\!80\)\( p^{25} T^{15} + 14635098273903618 p^{31} T^{16} + 18591100152028 p^{35} T^{17} + 1180446184 p^{40} T^{18} + 20496 p^{45} T^{19} + p^{50} T^{20} \)
47 \( 1 - 14970 T + 1168035829 T^{2} - 16946152573978 T^{3} + 759257371301274548 T^{4} - \)\(96\!\cdots\!06\)\( T^{5} + \)\(33\!\cdots\!87\)\( T^{6} - \)\(38\!\cdots\!86\)\( T^{7} + \)\(10\!\cdots\!71\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{9} + \)\(28\!\cdots\!68\)\( T^{10} - \)\(11\!\cdots\!72\)\( p^{5} T^{11} + \)\(10\!\cdots\!71\)\( p^{10} T^{12} - \)\(38\!\cdots\!86\)\( p^{15} T^{13} + \)\(33\!\cdots\!87\)\( p^{20} T^{14} - \)\(96\!\cdots\!06\)\( p^{25} T^{15} + 759257371301274548 p^{30} T^{16} - 16946152573978 p^{35} T^{17} + 1168035829 p^{40} T^{18} - 14970 p^{45} T^{19} + p^{50} T^{20} \)
53 \( 1 + 61980 T + 5238138690 T^{2} + 230304961147532 T^{3} + 11191476035686259205 T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(36\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} + \)\(43\!\cdots\!20\)\( p T^{9} + \)\(51\!\cdots\!96\)\( T^{10} + \)\(43\!\cdots\!20\)\( p^{6} T^{11} + \)\(10\!\cdots\!10\)\( p^{10} T^{12} + \)\(36\!\cdots\!12\)\( p^{15} T^{13} + \)\(13\!\cdots\!16\)\( p^{20} T^{14} + \)\(38\!\cdots\!20\)\( p^{25} T^{15} + 11191476035686259205 p^{30} T^{16} + 230304961147532 p^{35} T^{17} + 5238138690 p^{40} T^{18} + 61980 p^{45} T^{19} + p^{50} T^{20} \)
59 \( 1 - 61190 T + 4531368849 T^{2} - 185749072676294 T^{3} + 8992352159870334976 T^{4} - \)\(30\!\cdots\!74\)\( T^{5} + \)\(11\!\cdots\!27\)\( T^{6} - \)\(59\!\cdots\!02\)\( p T^{7} + \)\(12\!\cdots\!23\)\( T^{8} - \)\(32\!\cdots\!48\)\( T^{9} + \)\(97\!\cdots\!48\)\( T^{10} - \)\(32\!\cdots\!48\)\( p^{5} T^{11} + \)\(12\!\cdots\!23\)\( p^{10} T^{12} - \)\(59\!\cdots\!02\)\( p^{16} T^{13} + \)\(11\!\cdots\!27\)\( p^{20} T^{14} - \)\(30\!\cdots\!74\)\( p^{25} T^{15} + 8992352159870334976 p^{30} T^{16} - 185749072676294 p^{35} T^{17} + 4531368849 p^{40} T^{18} - 61190 p^{45} T^{19} + p^{50} T^{20} \)
61 \( 1 - 8230 T + 3297811083 T^{2} - 19489017366706 T^{3} + 5483173361138413117 T^{4} - \)\(35\!\cdots\!60\)\( T^{5} + \)\(69\!\cdots\!68\)\( T^{6} - \)\(44\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!82\)\( T^{8} - \)\(33\!\cdots\!04\)\( T^{9} + \)\(68\!\cdots\!54\)\( T^{10} - \)\(33\!\cdots\!04\)\( p^{5} T^{11} + \)\(75\!\cdots\!82\)\( p^{10} T^{12} - \)\(44\!\cdots\!28\)\( p^{15} T^{13} + \)\(69\!\cdots\!68\)\( p^{20} T^{14} - \)\(35\!\cdots\!60\)\( p^{25} T^{15} + 5483173361138413117 p^{30} T^{16} - 19489017366706 p^{35} T^{17} + 3297811083 p^{40} T^{18} - 8230 p^{45} T^{19} + p^{50} T^{20} \)
67 \( 1 - 11930 T + 8013703879 T^{2} - 134724221760354 T^{3} + 32899032963176424549 T^{4} - \)\(60\!\cdots\!24\)\( T^{5} + \)\(91\!\cdots\!84\)\( T^{6} - \)\(16\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!34\)\( T^{8} - \)\(30\!\cdots\!60\)\( T^{9} + \)\(28\!\cdots\!82\)\( T^{10} - \)\(30\!\cdots\!60\)\( p^{5} T^{11} + \)\(18\!\cdots\!34\)\( p^{10} T^{12} - \)\(16\!\cdots\!80\)\( p^{15} T^{13} + \)\(91\!\cdots\!84\)\( p^{20} T^{14} - \)\(60\!\cdots\!24\)\( p^{25} T^{15} + 32899032963176424549 p^{30} T^{16} - 134724221760354 p^{35} T^{17} + 8013703879 p^{40} T^{18} - 11930 p^{45} T^{19} + p^{50} T^{20} \)
71 \( 1 - 59822 T + 12713598104 T^{2} - 585604283801968 T^{3} + 74752525165203803761 T^{4} - \)\(28\!\cdots\!96\)\( T^{5} + \)\(28\!\cdots\!14\)\( T^{6} - \)\(93\!\cdots\!46\)\( T^{7} + \)\(77\!\cdots\!97\)\( T^{8} - \)\(22\!\cdots\!36\)\( T^{9} + \)\(15\!\cdots\!82\)\( T^{10} - \)\(22\!\cdots\!36\)\( p^{5} T^{11} + \)\(77\!\cdots\!97\)\( p^{10} T^{12} - \)\(93\!\cdots\!46\)\( p^{15} T^{13} + \)\(28\!\cdots\!14\)\( p^{20} T^{14} - \)\(28\!\cdots\!96\)\( p^{25} T^{15} + 74752525165203803761 p^{30} T^{16} - 585604283801968 p^{35} T^{17} + 12713598104 p^{40} T^{18} - 59822 p^{45} T^{19} + p^{50} T^{20} \)
73 \( 1 - 20680 T + 9699842910 T^{2} - 329929468103016 T^{3} + 53486424483246018493 T^{4} - \)\(21\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} - \)\(86\!\cdots\!68\)\( T^{7} + \)\(62\!\cdots\!66\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{9} + \)\(14\!\cdots\!64\)\( T^{10} - \)\(24\!\cdots\!92\)\( p^{5} T^{11} + \)\(62\!\cdots\!66\)\( p^{10} T^{12} - \)\(86\!\cdots\!68\)\( p^{15} T^{13} + \)\(20\!\cdots\!08\)\( p^{20} T^{14} - \)\(21\!\cdots\!40\)\( p^{25} T^{15} + 53486424483246018493 p^{30} T^{16} - 329929468103016 p^{35} T^{17} + 9699842910 p^{40} T^{18} - 20680 p^{45} T^{19} + p^{50} T^{20} \)
79 \( 1 - 234494 T + 46513402567 T^{2} - 6272183732333602 T^{3} + \)\(75\!\cdots\!60\)\( T^{4} - \)\(73\!\cdots\!78\)\( T^{5} + \)\(65\!\cdots\!65\)\( T^{6} - \)\(50\!\cdots\!38\)\( T^{7} + \)\(36\!\cdots\!67\)\( T^{8} - \)\(22\!\cdots\!76\)\( T^{9} + \)\(13\!\cdots\!80\)\( T^{10} - \)\(22\!\cdots\!76\)\( p^{5} T^{11} + \)\(36\!\cdots\!67\)\( p^{10} T^{12} - \)\(50\!\cdots\!38\)\( p^{15} T^{13} + \)\(65\!\cdots\!65\)\( p^{20} T^{14} - \)\(73\!\cdots\!78\)\( p^{25} T^{15} + \)\(75\!\cdots\!60\)\( p^{30} T^{16} - 6272183732333602 p^{35} T^{17} + 46513402567 p^{40} T^{18} - 234494 p^{45} T^{19} + p^{50} T^{20} \)
83 \( 1 + 185478 T + 34896456251 T^{2} + 4282063568026038 T^{3} + \)\(51\!\cdots\!09\)\( T^{4} + \)\(49\!\cdots\!36\)\( T^{5} + \)\(46\!\cdots\!68\)\( T^{6} + \)\(37\!\cdots\!28\)\( T^{7} + \)\(28\!\cdots\!10\)\( T^{8} + \)\(19\!\cdots\!52\)\( T^{9} + \)\(13\!\cdots\!30\)\( T^{10} + \)\(19\!\cdots\!52\)\( p^{5} T^{11} + \)\(28\!\cdots\!10\)\( p^{10} T^{12} + \)\(37\!\cdots\!28\)\( p^{15} T^{13} + \)\(46\!\cdots\!68\)\( p^{20} T^{14} + \)\(49\!\cdots\!36\)\( p^{25} T^{15} + \)\(51\!\cdots\!09\)\( p^{30} T^{16} + 4282063568026038 p^{35} T^{17} + 34896456251 p^{40} T^{18} + 185478 p^{45} T^{19} + p^{50} T^{20} \)
89 \( 1 - 181834 T + 29952755267 T^{2} - 3050003221466730 T^{3} + \)\(31\!\cdots\!89\)\( T^{4} - \)\(23\!\cdots\!84\)\( T^{5} + \)\(21\!\cdots\!64\)\( T^{6} - \)\(16\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(11\!\cdots\!58\)\( p T^{10} - \)\(12\!\cdots\!84\)\( p^{5} T^{11} + \)\(15\!\cdots\!98\)\( p^{10} T^{12} - \)\(16\!\cdots\!40\)\( p^{15} T^{13} + \)\(21\!\cdots\!64\)\( p^{20} T^{14} - \)\(23\!\cdots\!84\)\( p^{25} T^{15} + \)\(31\!\cdots\!89\)\( p^{30} T^{16} - 3050003221466730 p^{35} T^{17} + 29952755267 p^{40} T^{18} - 181834 p^{45} T^{19} + p^{50} T^{20} \)
97 \( 1 + 304358 T + 71052508415 T^{2} + 10598086203494238 T^{3} + \)\(13\!\cdots\!43\)\( T^{4} + \)\(12\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} + \)\(11\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!21\)\( T^{10} + \)\(11\!\cdots\!94\)\( p^{5} T^{11} + \)\(11\!\cdots\!97\)\( p^{10} T^{12} + \)\(10\!\cdots\!36\)\( p^{15} T^{13} + \)\(11\!\cdots\!30\)\( p^{20} T^{14} + \)\(12\!\cdots\!24\)\( p^{25} T^{15} + \)\(13\!\cdots\!43\)\( p^{30} T^{16} + 10598086203494238 p^{35} T^{17} + 71052508415 p^{40} T^{18} + 304358 p^{45} T^{19} + p^{50} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.16434221101772473507993529858, −3.02765840735965265349747479827, −2.63940337959941905672016699196, −2.60870248003433204662357243770, −2.42973247765588690170369427136, −2.40491554203558628746473248747, −2.10754842471413341230367386340, −2.08597934501081887978949251948, −1.85891838698969048187302143780, −1.58725250934802393415173784389, −1.57980249301412511509472623355, −1.54861738227629546742183803823, −1.36491137688305928849167539966, −1.36231460622469525076075752786, −1.32160035341704285520744341609, −1.17104405047908976279181639084, −0.847923877932123714902892218701, −0.74632116432464308477409735919, −0.67071606108734986708318034177, −0.62647572522832021175549768951, −0.60045209613967749526446907242, −0.41341405238948625160809312714, −0.39814640627144420110330139099, −0.29640139837403879074750780373, −0.02725736781901749989292177856, 0.02725736781901749989292177856, 0.29640139837403879074750780373, 0.39814640627144420110330139099, 0.41341405238948625160809312714, 0.60045209613967749526446907242, 0.62647572522832021175549768951, 0.67071606108734986708318034177, 0.74632116432464308477409735919, 0.847923877932123714902892218701, 1.17104405047908976279181639084, 1.32160035341704285520744341609, 1.36231460622469525076075752786, 1.36491137688305928849167539966, 1.54861738227629546742183803823, 1.57980249301412511509472623355, 1.58725250934802393415173784389, 1.85891838698969048187302143780, 2.08597934501081887978949251948, 2.10754842471413341230367386340, 2.40491554203558628746473248747, 2.42973247765588690170369427136, 2.60870248003433204662357243770, 2.63940337959941905672016699196, 3.02765840735965265349747479827, 3.16434221101772473507993529858

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

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