Properties

Degree 20
Conductor $ 43^{10} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 28·3-s − 27·4-s + 138·5-s + 224·6-s + 60·7-s − 374·8-s − 145·9-s + 1.10e3·10-s + 745·11-s − 756·12-s + 1.91e3·13-s + 480·14-s + 3.86e3·15-s − 153·16-s + 4.01e3·17-s − 1.16e3·18-s − 2.40e3·19-s − 3.72e3·20-s + 1.68e3·21-s + 5.96e3·22-s + 1.73e3·23-s − 1.04e4·24-s − 2.54e3·25-s + 1.53e4·26-s − 1.44e4·27-s − 1.62e3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.79·3-s − 0.843·4-s + 2.46·5-s + 2.54·6-s + 0.462·7-s − 2.06·8-s − 0.596·9-s + 3.49·10-s + 1.85·11-s − 1.51·12-s + 3.14·13-s + 0.654·14-s + 4.43·15-s − 0.149·16-s + 3.37·17-s − 0.843·18-s − 1.52·19-s − 2.08·20-s + 0.831·21-s + 2.62·22-s + 0.683·23-s − 3.71·24-s − 0.813·25-s + 4.44·26-s − 3.80·27-s − 0.390·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{10}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(43^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((20,\ 43^{10} ,\ ( \ : [5/2]^{10} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(68.5774\)
\(L(\frac12)\)  \(\approx\)  \(68.5774\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 20. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad43 \( ( 1 - p^{2} T )^{10} \)
good2 \( 1 - p^{3} T + 91 T^{2} - 285 p T^{3} + 2089 p T^{4} - 4803 p^{2} T^{5} + 31575 p^{2} T^{6} - 75439 p^{3} T^{7} + 290961 p^{4} T^{8} - 759243 p^{5} T^{9} + 1320203 p^{7} T^{10} - 759243 p^{10} T^{11} + 290961 p^{14} T^{12} - 75439 p^{18} T^{13} + 31575 p^{22} T^{14} - 4803 p^{27} T^{15} + 2089 p^{31} T^{16} - 285 p^{36} T^{17} + 91 p^{40} T^{18} - p^{48} T^{19} + p^{50} T^{20} \)
3 \( 1 - 28 T + 929 T^{2} - 15652 T^{3} + 384094 T^{4} - 1901098 p T^{5} + 4711762 p^{3} T^{6} - 65694782 p^{3} T^{7} + 157703879 p^{5} T^{8} - 2106657868 p^{5} T^{9} + 13931284666 p^{6} T^{10} - 2106657868 p^{10} T^{11} + 157703879 p^{15} T^{12} - 65694782 p^{18} T^{13} + 4711762 p^{23} T^{14} - 1901098 p^{26} T^{15} + 384094 p^{30} T^{16} - 15652 p^{35} T^{17} + 929 p^{40} T^{18} - 28 p^{45} T^{19} + p^{50} T^{20} \)
5 \( 1 - 138 T + 21587 T^{2} - 2230608 T^{3} + 229459202 T^{4} - 3831534036 p T^{5} + 1558167162738 T^{6} - 109829488059312 T^{7} + 7514260269046033 T^{8} - 457924168440376602 T^{9} + 27012319396517106654 T^{10} - 457924168440376602 p^{5} T^{11} + 7514260269046033 p^{10} T^{12} - 109829488059312 p^{15} T^{13} + 1558167162738 p^{20} T^{14} - 3831534036 p^{26} T^{15} + 229459202 p^{30} T^{16} - 2230608 p^{35} T^{17} + 21587 p^{40} T^{18} - 138 p^{45} T^{19} + p^{50} T^{20} \)
7 \( 1 - 60 T + 10158 p T^{2} - 810396 p T^{3} + 2998880769 T^{4} - 247503266272 T^{5} + 92903828588736 T^{6} - 7232195163049824 T^{7} + 2203025939434531374 T^{8} - \)\(15\!\cdots\!56\)\( T^{9} + \)\(41\!\cdots\!04\)\( T^{10} - \)\(15\!\cdots\!56\)\( p^{5} T^{11} + 2203025939434531374 p^{10} T^{12} - 7232195163049824 p^{15} T^{13} + 92903828588736 p^{20} T^{14} - 247503266272 p^{25} T^{15} + 2998880769 p^{30} T^{16} - 810396 p^{36} T^{17} + 10158 p^{41} T^{18} - 60 p^{45} T^{19} + p^{50} T^{20} \)
11 \( 1 - 745 T + 1092200 T^{2} - 51526415 p T^{3} + 495359905314 T^{4} - 194173124255379 T^{5} + 135383021483146674 T^{6} - 356187389061889985 p^{2} T^{7} + \)\(27\!\cdots\!81\)\( T^{8} - \)\(77\!\cdots\!58\)\( T^{9} + \)\(43\!\cdots\!56\)\( p T^{10} - \)\(77\!\cdots\!58\)\( p^{5} T^{11} + \)\(27\!\cdots\!81\)\( p^{10} T^{12} - 356187389061889985 p^{17} T^{13} + 135383021483146674 p^{20} T^{14} - 194173124255379 p^{25} T^{15} + 495359905314 p^{30} T^{16} - 51526415 p^{36} T^{17} + 1092200 p^{40} T^{18} - 745 p^{45} T^{19} + p^{50} T^{20} \)
13 \( 1 - 1917 T + 3889346 T^{2} - 4848596699 T^{3} + 5963954609742 T^{4} - 5730569373162787 T^{5} + 5360926546177278684 T^{6} - \)\(42\!\cdots\!79\)\( T^{7} + \)\(32\!\cdots\!21\)\( T^{8} - \)\(21\!\cdots\!94\)\( T^{9} + \)\(14\!\cdots\!96\)\( T^{10} - \)\(21\!\cdots\!94\)\( p^{5} T^{11} + \)\(32\!\cdots\!21\)\( p^{10} T^{12} - \)\(42\!\cdots\!79\)\( p^{15} T^{13} + 5360926546177278684 p^{20} T^{14} - 5730569373162787 p^{25} T^{15} + 5963954609742 p^{30} T^{16} - 4848596699 p^{35} T^{17} + 3889346 p^{40} T^{18} - 1917 p^{45} T^{19} + p^{50} T^{20} \)
17 \( 1 - 4017 T + 16731071 T^{2} - 43371481422 T^{3} + 109028323359989 T^{4} - 214433083727073801 T^{5} + \)\(40\!\cdots\!11\)\( T^{6} - \)\(64\!\cdots\!84\)\( T^{7} + \)\(98\!\cdots\!87\)\( T^{8} - \)\(13\!\cdots\!91\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(13\!\cdots\!91\)\( p^{5} T^{11} + \)\(98\!\cdots\!87\)\( p^{10} T^{12} - \)\(64\!\cdots\!84\)\( p^{15} T^{13} + \)\(40\!\cdots\!11\)\( p^{20} T^{14} - 214433083727073801 p^{25} T^{15} + 109028323359989 p^{30} T^{16} - 43371481422 p^{35} T^{17} + 16731071 p^{40} T^{18} - 4017 p^{45} T^{19} + p^{50} T^{20} \)
19 \( 1 + 2404 T + 15222557 T^{2} + 35388594664 T^{3} + 123780260921510 T^{4} + 260083719347815286 T^{5} + \)\(66\!\cdots\!54\)\( T^{6} + \)\(12\!\cdots\!42\)\( T^{7} + \)\(25\!\cdots\!37\)\( T^{8} + \)\(41\!\cdots\!16\)\( T^{9} + \)\(73\!\cdots\!66\)\( T^{10} + \)\(41\!\cdots\!16\)\( p^{5} T^{11} + \)\(25\!\cdots\!37\)\( p^{10} T^{12} + \)\(12\!\cdots\!42\)\( p^{15} T^{13} + \)\(66\!\cdots\!54\)\( p^{20} T^{14} + 260083719347815286 p^{25} T^{15} + 123780260921510 p^{30} T^{16} + 35388594664 p^{35} T^{17} + 15222557 p^{40} T^{18} + 2404 p^{45} T^{19} + p^{50} T^{20} \)
23 \( 1 - 1733 T + 47557957 T^{2} - 72067471074 T^{3} + 1063199488621929 T^{4} - 1395811354288602239 T^{5} + \)\(14\!\cdots\!11\)\( T^{6} - \)\(16\!\cdots\!50\)\( T^{7} + \)\(64\!\cdots\!93\)\( p T^{8} - \)\(14\!\cdots\!37\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} - \)\(14\!\cdots\!37\)\( p^{5} T^{11} + \)\(64\!\cdots\!93\)\( p^{11} T^{12} - \)\(16\!\cdots\!50\)\( p^{15} T^{13} + \)\(14\!\cdots\!11\)\( p^{20} T^{14} - 1395811354288602239 p^{25} T^{15} + 1063199488621929 p^{30} T^{16} - 72067471074 p^{35} T^{17} + 47557957 p^{40} T^{18} - 1733 p^{45} T^{19} + p^{50} T^{20} \)
29 \( 1 - 6996 T + 88174667 T^{2} - 354425388876 T^{3} + 2685110189553942 T^{4} - 3595496283776034360 T^{5} + \)\(29\!\cdots\!74\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} - \)\(13\!\cdots\!47\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{9} - \)\(82\!\cdots\!38\)\( T^{10} + \)\(63\!\cdots\!00\)\( p^{5} T^{11} - \)\(13\!\cdots\!47\)\( p^{10} T^{12} + \)\(15\!\cdots\!88\)\( p^{15} T^{13} + \)\(29\!\cdots\!74\)\( p^{20} T^{14} - 3595496283776034360 p^{25} T^{15} + 2685110189553942 p^{30} T^{16} - 354425388876 p^{35} T^{17} + 88174667 p^{40} T^{18} - 6996 p^{45} T^{19} + p^{50} T^{20} \)
31 \( 1 + 4899 T + 169335305 T^{2} + 687234491898 T^{3} + 14314848605964873 T^{4} + 50115254632047551877 T^{5} + \)\(80\!\cdots\!39\)\( T^{6} + \)\(24\!\cdots\!98\)\( T^{7} + \)\(33\!\cdots\!11\)\( T^{8} + \)\(91\!\cdots\!15\)\( T^{9} + \)\(10\!\cdots\!46\)\( T^{10} + \)\(91\!\cdots\!15\)\( p^{5} T^{11} + \)\(33\!\cdots\!11\)\( p^{10} T^{12} + \)\(24\!\cdots\!98\)\( p^{15} T^{13} + \)\(80\!\cdots\!39\)\( p^{20} T^{14} + 50115254632047551877 p^{25} T^{15} + 14314848605964873 p^{30} T^{16} + 687234491898 p^{35} T^{17} + 169335305 p^{40} T^{18} + 4899 p^{45} T^{19} + p^{50} T^{20} \)
37 \( 1 - 1466 T + 314595515 T^{2} - 1000159162152 T^{3} + 48977417173952218 T^{4} - \)\(27\!\cdots\!48\)\( T^{5} + \)\(50\!\cdots\!42\)\( T^{6} - \)\(41\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} - \)\(41\!\cdots\!94\)\( T^{9} + \)\(29\!\cdots\!30\)\( T^{10} - \)\(41\!\cdots\!94\)\( p^{5} T^{11} + \)\(40\!\cdots\!01\)\( p^{10} T^{12} - \)\(41\!\cdots\!16\)\( p^{15} T^{13} + \)\(50\!\cdots\!42\)\( p^{20} T^{14} - \)\(27\!\cdots\!48\)\( p^{25} T^{15} + 48977417173952218 p^{30} T^{16} - 1000159162152 p^{35} T^{17} + 314595515 p^{40} T^{18} - 1466 p^{45} T^{19} + p^{50} T^{20} \)
41 \( 1 - 10297 T + 512459663 T^{2} - 5996761727014 T^{3} + 151679237600092005 T^{4} - \)\(17\!\cdots\!49\)\( T^{5} + \)\(31\!\cdots\!39\)\( T^{6} - \)\(34\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!19\)\( T^{8} - \)\(50\!\cdots\!39\)\( T^{9} + \)\(65\!\cdots\!96\)\( T^{10} - \)\(50\!\cdots\!39\)\( p^{5} T^{11} + \)\(50\!\cdots\!19\)\( p^{10} T^{12} - \)\(34\!\cdots\!68\)\( p^{15} T^{13} + \)\(31\!\cdots\!39\)\( p^{20} T^{14} - \)\(17\!\cdots\!49\)\( p^{25} T^{15} + 151679237600092005 p^{30} T^{16} - 5996761727014 p^{35} T^{17} + 512459663 p^{40} T^{18} - 10297 p^{45} T^{19} + p^{50} T^{20} \)
47 \( 1 - 48592 T + 2515660075 T^{2} - 84810946501416 T^{3} + 2666904651640316270 T^{4} - \)\(68\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!42\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + \)\(64\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!66\)\( T^{10} - \)\(11\!\cdots\!92\)\( p^{5} T^{11} + \)\(64\!\cdots\!57\)\( p^{10} T^{12} - \)\(33\!\cdots\!20\)\( p^{15} T^{13} + \)\(16\!\cdots\!42\)\( p^{20} T^{14} - \)\(68\!\cdots\!52\)\( p^{25} T^{15} + 2666904651640316270 p^{30} T^{16} - 84810946501416 p^{35} T^{17} + 2515660075 p^{40} T^{18} - 48592 p^{45} T^{19} + p^{50} T^{20} \)
53 \( 1 - 127165 T + 9906770390 T^{2} - 550920294329671 T^{3} + 24581308628857110150 T^{4} - \)\(91\!\cdots\!35\)\( T^{5} + \)\(29\!\cdots\!20\)\( T^{6} - \)\(83\!\cdots\!31\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(49\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} - \)\(49\!\cdots\!78\)\( p^{5} T^{11} + \)\(21\!\cdots\!05\)\( p^{10} T^{12} - \)\(83\!\cdots\!31\)\( p^{15} T^{13} + \)\(29\!\cdots\!20\)\( p^{20} T^{14} - \)\(91\!\cdots\!35\)\( p^{25} T^{15} + 24581308628857110150 p^{30} T^{16} - 550920294329671 p^{35} T^{17} + 9906770390 p^{40} T^{18} - 127165 p^{45} T^{19} + p^{50} T^{20} \)
59 \( 1 - 99372 T + 7563799766 T^{2} - 407898209580180 T^{3} + 19000827281933813957 T^{4} - \)\(74\!\cdots\!72\)\( T^{5} + \)\(26\!\cdots\!44\)\( T^{6} - \)\(87\!\cdots\!68\)\( T^{7} + \)\(26\!\cdots\!22\)\( T^{8} - \)\(76\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!96\)\( T^{10} - \)\(76\!\cdots\!76\)\( p^{5} T^{11} + \)\(26\!\cdots\!22\)\( p^{10} T^{12} - \)\(87\!\cdots\!68\)\( p^{15} T^{13} + \)\(26\!\cdots\!44\)\( p^{20} T^{14} - \)\(74\!\cdots\!72\)\( p^{25} T^{15} + 19000827281933813957 p^{30} T^{16} - 407898209580180 p^{35} T^{17} + 7563799766 p^{40} T^{18} - 99372 p^{45} T^{19} + p^{50} T^{20} \)
61 \( 1 - 17408 T + 3107203610 T^{2} - 59824183442672 T^{3} + 5726081070022748105 T^{4} - \)\(10\!\cdots\!64\)\( T^{5} + \)\(76\!\cdots\!32\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(84\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!80\)\( T^{9} + \)\(77\!\cdots\!00\)\( T^{10} - \)\(13\!\cdots\!80\)\( p^{5} T^{11} + \)\(84\!\cdots\!70\)\( p^{10} T^{12} - \)\(13\!\cdots\!44\)\( p^{15} T^{13} + \)\(76\!\cdots\!32\)\( p^{20} T^{14} - \)\(10\!\cdots\!64\)\( p^{25} T^{15} + 5726081070022748105 p^{30} T^{16} - 59824183442672 p^{35} T^{17} + 3107203610 p^{40} T^{18} - 17408 p^{45} T^{19} + p^{50} T^{20} \)
67 \( 1 + 2021 T + 3389673908 T^{2} - 95129046044495 T^{3} + 5951903574650911354 T^{4} - \)\(26\!\cdots\!45\)\( T^{5} + \)\(15\!\cdots\!46\)\( T^{6} - \)\(38\!\cdots\!75\)\( T^{7} + \)\(27\!\cdots\!93\)\( T^{8} - \)\(78\!\cdots\!74\)\( T^{9} + \)\(36\!\cdots\!28\)\( T^{10} - \)\(78\!\cdots\!74\)\( p^{5} T^{11} + \)\(27\!\cdots\!93\)\( p^{10} T^{12} - \)\(38\!\cdots\!75\)\( p^{15} T^{13} + \)\(15\!\cdots\!46\)\( p^{20} T^{14} - \)\(26\!\cdots\!45\)\( p^{25} T^{15} + 5951903574650911354 p^{30} T^{16} - 95129046044495 p^{35} T^{17} + 3389673908 p^{40} T^{18} + 2021 p^{45} T^{19} + p^{50} T^{20} \)
71 \( 1 - 11286 T + 12540359910 T^{2} - 44404438984650 T^{3} + 71277357843109037437 T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(24\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(61\!\cdots\!22\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!84\)\( T^{10} + \)\(10\!\cdots\!68\)\( p^{5} T^{11} + \)\(61\!\cdots\!22\)\( p^{10} T^{12} + \)\(30\!\cdots\!96\)\( p^{15} T^{13} + \)\(24\!\cdots\!24\)\( p^{20} T^{14} + \)\(33\!\cdots\!36\)\( p^{25} T^{15} + 71277357843109037437 p^{30} T^{16} - 44404438984650 p^{35} T^{17} + 12540359910 p^{40} T^{18} - 11286 p^{45} T^{19} + p^{50} T^{20} \)
73 \( 1 - 49892 T + 9676771082 T^{2} - 435365481214596 T^{3} + 51788415419782211185 T^{4} - \)\(30\!\cdots\!56\)\( p T^{5} + \)\(19\!\cdots\!88\)\( T^{6} - \)\(79\!\cdots\!24\)\( T^{7} + \)\(57\!\cdots\!14\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{9} + \)\(13\!\cdots\!60\)\( T^{10} - \)\(21\!\cdots\!32\)\( p^{5} T^{11} + \)\(57\!\cdots\!14\)\( p^{10} T^{12} - \)\(79\!\cdots\!24\)\( p^{15} T^{13} + \)\(19\!\cdots\!88\)\( p^{20} T^{14} - \)\(30\!\cdots\!56\)\( p^{26} T^{15} + 51788415419782211185 p^{30} T^{16} - 435365481214596 p^{35} T^{17} + 9676771082 p^{40} T^{18} - 49892 p^{45} T^{19} + p^{50} T^{20} \)
79 \( 1 + 91524 T + 16857154847 T^{2} + 842558519651240 T^{3} + 99200238774687370926 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(59\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!17\)\( T^{8} + \)\(38\!\cdots\!04\)\( T^{9} + \)\(63\!\cdots\!18\)\( T^{10} + \)\(38\!\cdots\!04\)\( p^{5} T^{11} + \)\(16\!\cdots\!17\)\( p^{10} T^{12} + \)\(59\!\cdots\!56\)\( p^{15} T^{13} + \)\(37\!\cdots\!90\)\( p^{20} T^{14} + \)\(23\!\cdots\!36\)\( p^{25} T^{15} + 99200238774687370926 p^{30} T^{16} + 842558519651240 p^{35} T^{17} + 16857154847 p^{40} T^{18} + 91524 p^{45} T^{19} + p^{50} T^{20} \)
83 \( 1 + 105203 T + 28621889044 T^{2} + 2443253098090515 T^{3} + \)\(38\!\cdots\!90\)\( T^{4} + \)\(27\!\cdots\!61\)\( T^{5} + \)\(31\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{7} + \)\(18\!\cdots\!57\)\( T^{8} + \)\(99\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!32\)\( T^{10} + \)\(99\!\cdots\!22\)\( p^{5} T^{11} + \)\(18\!\cdots\!57\)\( p^{10} T^{12} + \)\(19\!\cdots\!11\)\( p^{15} T^{13} + \)\(31\!\cdots\!66\)\( p^{20} T^{14} + \)\(27\!\cdots\!61\)\( p^{25} T^{15} + \)\(38\!\cdots\!90\)\( p^{30} T^{16} + 2443253098090515 p^{35} T^{17} + 28621889044 p^{40} T^{18} + 105203 p^{45} T^{19} + p^{50} T^{20} \)
89 \( 1 + 62682 T + 34400806898 T^{2} + 1591902274833594 T^{3} + \)\(57\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!96\)\( T^{5} + \)\(63\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!12\)\( T^{7} + \)\(50\!\cdots\!94\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(31\!\cdots\!32\)\( T^{10} + \)\(10\!\cdots\!68\)\( p^{5} T^{11} + \)\(50\!\cdots\!94\)\( p^{10} T^{12} + \)\(16\!\cdots\!12\)\( p^{15} T^{13} + \)\(63\!\cdots\!68\)\( p^{20} T^{14} + \)\(20\!\cdots\!96\)\( p^{25} T^{15} + \)\(57\!\cdots\!17\)\( p^{30} T^{16} + 1591902274833594 p^{35} T^{17} + 34400806898 p^{40} T^{18} + 62682 p^{45} T^{19} + p^{50} T^{20} \)
97 \( 1 - 108383 T + 55022053811 T^{2} - 4727913772355050 T^{3} + \)\(13\!\cdots\!17\)\( T^{4} - \)\(99\!\cdots\!31\)\( T^{5} + \)\(22\!\cdots\!35\)\( T^{6} - \)\(13\!\cdots\!96\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(14\!\cdots\!33\)\( T^{9} + \)\(25\!\cdots\!28\)\( T^{10} - \)\(14\!\cdots\!33\)\( p^{5} T^{11} + \)\(26\!\cdots\!83\)\( p^{10} T^{12} - \)\(13\!\cdots\!96\)\( p^{15} T^{13} + \)\(22\!\cdots\!35\)\( p^{20} T^{14} - \)\(99\!\cdots\!31\)\( p^{25} T^{15} + \)\(13\!\cdots\!17\)\( p^{30} T^{16} - 4727913772355050 p^{35} T^{17} + 55022053811 p^{40} T^{18} - 108383 p^{45} T^{19} + p^{50} T^{20} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.62411278443316773569213887066, −5.43434478862698674201748772362, −5.38701354559212273686749777793, −5.24867699807912780449493787938, −5.15950162687004251845581096357, −4.49823628971596812822810618290, −4.31147431097332069979659576886, −4.30807811987455266889374395728, −4.11832708096723618849682078976, −3.84167185072649783030299320982, −3.69938247848136015406298073891, −3.66461940834268703106483122238, −3.55919740872722686252924413259, −3.39043826316152718059482802822, −2.91965384678829062864018954373, −2.68021637322697675678592702317, −2.44971709892072270243620974604, −2.28343938478795840589969228638, −2.15692329361905417967922491558, −1.69946400093387293691053616524, −1.53976195818631881285740316936, −1.16736011070674858823024433785, −0.892984695343857725849962469810, −0.872117925969098084697913757559, −0.35875812059184369204644146503, 0.35875812059184369204644146503, 0.872117925969098084697913757559, 0.892984695343857725849962469810, 1.16736011070674858823024433785, 1.53976195818631881285740316936, 1.69946400093387293691053616524, 2.15692329361905417967922491558, 2.28343938478795840589969228638, 2.44971709892072270243620974604, 2.68021637322697675678592702317, 2.91965384678829062864018954373, 3.39043826316152718059482802822, 3.55919740872722686252924413259, 3.66461940834268703106483122238, 3.69938247848136015406298073891, 3.84167185072649783030299320982, 4.11832708096723618849682078976, 4.30807811987455266889374395728, 4.31147431097332069979659576886, 4.49823628971596812822810618290, 5.15950162687004251845581096357, 5.24867699807912780449493787938, 5.38701354559212273686749777793, 5.43434478862698674201748772362, 5.62411278443316773569213887066

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

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