Properties

Label 40-384e20-1.1-c5e20-0-2
Degree $40$
Conductor $4.860\times 10^{51}$
Sign $1$
Analytic cond. $6.16361\times 10^{35}$
Root an. cond. $7.84776$
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  + 2·3-s + 2·9-s + 948·11-s − 328·23-s + 2.50e4·25-s + 678·27-s + 1.89e3·33-s + 1.50e4·37-s − 3.66e4·47-s + 1.51e5·49-s + 6.29e4·59-s + 7.32e4·61-s − 656·69-s − 3.48e4·71-s + 5.25e4·73-s + 5.00e4·75-s + 1.52e4·81-s − 2.25e5·83-s + 7.60e3·97-s + 1.89e3·99-s + 1.24e5·107-s − 1.73e5·109-s + 3.01e4·111-s − 1.06e6·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.128·3-s + 0.00823·9-s + 2.36·11-s − 0.129·23-s + 8·25-s + 0.178·27-s + 0.303·33-s + 1.80·37-s − 2.41·47-s + 9.00·49-s + 2.35·59-s + 2.52·61-s − 0.0165·69-s − 0.821·71-s + 1.15·73-s + 1.02·75-s + 0.257·81-s − 3.58·83-s + 0.0820·97-s + 0.0194·99-s + 1.04·107-s − 1.39·109-s + 0.231·111-s − 6.59·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{140} \cdot 3^{20}\)
Sign: $1$
Analytic conductor: \(6.16361\times 10^{35}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.01285038375\)
\(L(\frac12)\) \(\approx\) \(0.01285038375\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 2 T + 2 T^{2} - 226 p T^{3} - 4169 p T^{4} - 163832 p^{2} T^{5} - 92872 p^{2} T^{6} + 2714200 p^{4} T^{7} - 1259962 p^{7} T^{8} + 5104756 p^{8} T^{9} + 18110828 p^{10} T^{10} + 5104756 p^{13} T^{11} - 1259962 p^{17} T^{12} + 2714200 p^{19} T^{13} - 92872 p^{22} T^{14} - 163832 p^{27} T^{15} - 4169 p^{31} T^{16} - 226 p^{36} T^{17} + 2 p^{40} T^{18} - 2 p^{45} T^{19} + p^{50} T^{20} \)
good5 \( 1 - 8 p^{5} T^{2} + 64922862 p T^{4} - 581176159496 p T^{6} + 805161162779173 p^{2} T^{8} - \)\(11\!\cdots\!08\)\( T^{10} + \)\(44\!\cdots\!92\)\( p^{3} T^{12} - \)\(48\!\cdots\!28\)\( p T^{14} + \)\(18\!\cdots\!14\)\( p T^{16} - \)\(13\!\cdots\!24\)\( p^{2} T^{18} + \)\(10\!\cdots\!16\)\( T^{20} - \)\(13\!\cdots\!24\)\( p^{12} T^{22} + \)\(18\!\cdots\!14\)\( p^{21} T^{24} - \)\(48\!\cdots\!28\)\( p^{31} T^{26} + \)\(44\!\cdots\!92\)\( p^{43} T^{28} - \)\(11\!\cdots\!08\)\( p^{50} T^{30} + 805161162779173 p^{62} T^{32} - 581176159496 p^{71} T^{34} + 64922862 p^{81} T^{36} - 8 p^{95} T^{38} + p^{100} T^{40} \)
7 \( 1 - 151376 T^{2} + 11636394822 T^{4} - 602773429074896 T^{6} + 23724202393315072525 T^{8} - \)\(76\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!20\)\( T^{12} - \)\(50\!\cdots\!84\)\( T^{14} + \)\(15\!\cdots\!74\)\( p T^{16} - \)\(21\!\cdots\!52\)\( T^{18} + \)\(37\!\cdots\!28\)\( T^{20} - \)\(21\!\cdots\!52\)\( p^{10} T^{22} + \)\(15\!\cdots\!74\)\( p^{21} T^{24} - \)\(50\!\cdots\!84\)\( p^{30} T^{26} + \)\(20\!\cdots\!20\)\( p^{40} T^{28} - \)\(76\!\cdots\!68\)\( p^{50} T^{30} + 23724202393315072525 p^{60} T^{32} - 602773429074896 p^{70} T^{34} + 11636394822 p^{80} T^{36} - 151376 p^{90} T^{38} + p^{100} T^{40} \)
11 \( ( 1 - 474 T + 867706 T^{2} - 318904062 T^{3} + 2808086013 p^{2} T^{4} - 103919584780056 T^{5} + 85745787142006680 T^{6} - 23781595566239469960 T^{7} + \)\(16\!\cdots\!22\)\( T^{8} - \)\(44\!\cdots\!48\)\( T^{9} + \)\(29\!\cdots\!36\)\( T^{10} - \)\(44\!\cdots\!48\)\( p^{5} T^{11} + \)\(16\!\cdots\!22\)\( p^{10} T^{12} - 23781595566239469960 p^{15} T^{13} + 85745787142006680 p^{20} T^{14} - 103919584780056 p^{25} T^{15} + 2808086013 p^{32} T^{16} - 318904062 p^{35} T^{17} + 867706 p^{40} T^{18} - 474 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
13 \( ( 1 + 1727358 T^{2} + 389121024 T^{3} + 1484851902293 T^{4} + 633595156478976 T^{5} + 897326840696086824 T^{6} + \)\(51\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!66\)\( T^{8} + \)\(27\!\cdots\!56\)\( T^{9} + \)\(17\!\cdots\!76\)\( T^{10} + \)\(27\!\cdots\!56\)\( p^{5} T^{11} + \)\(43\!\cdots\!66\)\( p^{10} T^{12} + \)\(51\!\cdots\!00\)\( p^{15} T^{13} + 897326840696086824 p^{20} T^{14} + 633595156478976 p^{25} T^{15} + 1484851902293 p^{30} T^{16} + 389121024 p^{35} T^{17} + 1727358 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
17 \( 1 - 12402532 T^{2} + 4475762806206 p T^{4} - \)\(31\!\cdots\!88\)\( T^{6} + \)\(97\!\cdots\!17\)\( T^{8} - \)\(24\!\cdots\!20\)\( T^{10} + \)\(54\!\cdots\!88\)\( T^{12} - \)\(10\!\cdots\!08\)\( T^{14} + \)\(18\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!72\)\( T^{18} + \)\(44\!\cdots\!08\)\( T^{20} - \)\(30\!\cdots\!72\)\( p^{10} T^{22} + \)\(18\!\cdots\!38\)\( p^{20} T^{24} - \)\(10\!\cdots\!08\)\( p^{30} T^{26} + \)\(54\!\cdots\!88\)\( p^{40} T^{28} - \)\(24\!\cdots\!20\)\( p^{50} T^{30} + \)\(97\!\cdots\!17\)\( p^{60} T^{32} - \)\(31\!\cdots\!88\)\( p^{70} T^{34} + 4475762806206 p^{81} T^{36} - 12402532 p^{90} T^{38} + p^{100} T^{40} \)
19 \( 1 - 1217616 p T^{2} + 283813760957590 T^{4} - \)\(24\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!25\)\( T^{8} - \)\(82\!\cdots\!56\)\( T^{10} + \)\(36\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!24\)\( T^{14} + \)\(24\!\cdots\!14\)\( p T^{16} - \)\(13\!\cdots\!44\)\( T^{18} + \)\(10\!\cdots\!80\)\( p^{2} T^{20} - \)\(13\!\cdots\!44\)\( p^{10} T^{22} + \)\(24\!\cdots\!14\)\( p^{21} T^{24} - \)\(14\!\cdots\!24\)\( p^{30} T^{26} + \)\(36\!\cdots\!96\)\( p^{40} T^{28} - \)\(82\!\cdots\!56\)\( p^{50} T^{30} + \)\(15\!\cdots\!25\)\( p^{60} T^{32} - \)\(24\!\cdots\!04\)\( p^{70} T^{34} + 283813760957590 p^{80} T^{36} - 1217616 p^{91} T^{38} + p^{100} T^{40} \)
23 \( ( 1 + 164 T + 31861174 T^{2} + 8842821788 T^{3} + 496564216557245 T^{4} + 254302378049959056 T^{5} + \)\(52\!\cdots\!24\)\( T^{6} + \)\(39\!\cdots\!36\)\( T^{7} + \)\(44\!\cdots\!66\)\( T^{8} + \)\(38\!\cdots\!84\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} + \)\(38\!\cdots\!84\)\( p^{5} T^{11} + \)\(44\!\cdots\!66\)\( p^{10} T^{12} + \)\(39\!\cdots\!36\)\( p^{15} T^{13} + \)\(52\!\cdots\!24\)\( p^{20} T^{14} + 254302378049959056 p^{25} T^{15} + 496564216557245 p^{30} T^{16} + 8842821788 p^{35} T^{17} + 31861174 p^{40} T^{18} + 164 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
29 \( 1 - 191113384 T^{2} + 18110181894055494 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(55\!\cdots\!05\)\( T^{8} - \)\(22\!\cdots\!64\)\( T^{10} + \)\(76\!\cdots\!68\)\( T^{12} - \)\(23\!\cdots\!04\)\( T^{14} + \)\(62\!\cdots\!14\)\( T^{16} - \)\(14\!\cdots\!84\)\( T^{18} + \)\(32\!\cdots\!16\)\( T^{20} - \)\(14\!\cdots\!84\)\( p^{10} T^{22} + \)\(62\!\cdots\!14\)\( p^{20} T^{24} - \)\(23\!\cdots\!04\)\( p^{30} T^{26} + \)\(76\!\cdots\!68\)\( p^{40} T^{28} - \)\(22\!\cdots\!64\)\( p^{50} T^{30} + \)\(55\!\cdots\!05\)\( p^{60} T^{32} - \)\(11\!\cdots\!24\)\( p^{70} T^{34} + 18110181894055494 p^{80} T^{36} - 191113384 p^{90} T^{38} + p^{100} T^{40} \)
31 \( 1 - 252335328 T^{2} + 33282623187817222 T^{4} - \)\(30\!\cdots\!28\)\( T^{6} + \)\(21\!\cdots\!37\)\( T^{8} - \)\(12\!\cdots\!56\)\( T^{10} + \)\(63\!\cdots\!56\)\( T^{12} - \)\(27\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!66\)\( T^{16} - \)\(36\!\cdots\!04\)\( T^{18} + \)\(10\!\cdots\!36\)\( T^{20} - \)\(36\!\cdots\!04\)\( p^{10} T^{22} + \)\(10\!\cdots\!66\)\( p^{20} T^{24} - \)\(27\!\cdots\!84\)\( p^{30} T^{26} + \)\(63\!\cdots\!56\)\( p^{40} T^{28} - \)\(12\!\cdots\!56\)\( p^{50} T^{30} + \)\(21\!\cdots\!37\)\( p^{60} T^{32} - \)\(30\!\cdots\!28\)\( p^{70} T^{34} + 33282623187817222 p^{80} T^{36} - 252335328 p^{90} T^{38} + p^{100} T^{40} \)
37 \( ( 1 - 7528 T + 415621758 T^{2} - 2534251060552 T^{3} + 86513533821089413 T^{4} - \)\(44\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} - \)\(52\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!22\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{9} + \)\(94\!\cdots\!20\)\( T^{10} - \)\(46\!\cdots\!60\)\( p^{5} T^{11} + \)\(12\!\cdots\!22\)\( p^{10} T^{12} - \)\(52\!\cdots\!20\)\( p^{15} T^{13} + \)\(11\!\cdots\!68\)\( p^{20} T^{14} - \)\(44\!\cdots\!08\)\( p^{25} T^{15} + 86513533821089413 p^{30} T^{16} - 2534251060552 p^{35} T^{17} + 415621758 p^{40} T^{18} - 7528 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
41 \( 1 - 992892100 T^{2} + 507553214877646398 T^{4} - \)\(17\!\cdots\!64\)\( T^{6} + \)\(48\!\cdots\!93\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(20\!\cdots\!48\)\( T^{12} - \)\(33\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!10\)\( T^{16} - \)\(65\!\cdots\!64\)\( T^{18} + \)\(78\!\cdots\!68\)\( T^{20} - \)\(65\!\cdots\!64\)\( p^{10} T^{22} + \)\(49\!\cdots\!10\)\( p^{20} T^{24} - \)\(33\!\cdots\!08\)\( p^{30} T^{26} + \)\(20\!\cdots\!48\)\( p^{40} T^{28} - \)\(10\!\cdots\!80\)\( p^{50} T^{30} + \)\(48\!\cdots\!93\)\( p^{60} T^{32} - \)\(17\!\cdots\!64\)\( p^{70} T^{34} + 507553214877646398 p^{80} T^{36} - 992892100 p^{90} T^{38} + p^{100} T^{40} \)
43 \( 1 - 1462691120 T^{2} + 1076203296774082038 T^{4} - \)\(52\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!73\)\( T^{8} - \)\(56\!\cdots\!40\)\( T^{10} + \)\(14\!\cdots\!92\)\( T^{12} - \)\(30\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!70\)\( T^{16} - \)\(99\!\cdots\!88\)\( T^{18} + \)\(15\!\cdots\!12\)\( T^{20} - \)\(99\!\cdots\!88\)\( p^{10} T^{22} + \)\(57\!\cdots\!70\)\( p^{20} T^{24} - \)\(30\!\cdots\!84\)\( p^{30} T^{26} + \)\(14\!\cdots\!92\)\( p^{40} T^{28} - \)\(56\!\cdots\!40\)\( p^{50} T^{30} + \)\(19\!\cdots\!73\)\( p^{60} T^{32} - \)\(52\!\cdots\!20\)\( p^{70} T^{34} + 1076203296774082038 p^{80} T^{36} - 1462691120 p^{90} T^{38} + p^{100} T^{40} \)
47 \( ( 1 + 18320 T + 1471207574 T^{2} + 19754962156272 T^{3} + 989833625114591949 T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(43\!\cdots\!60\)\( T^{6} + \)\(38\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!78\)\( T^{8} + \)\(11\!\cdots\!12\)\( T^{9} + \)\(37\!\cdots\!12\)\( T^{10} + \)\(11\!\cdots\!12\)\( p^{5} T^{11} + \)\(14\!\cdots\!78\)\( p^{10} T^{12} + \)\(38\!\cdots\!40\)\( p^{15} T^{13} + \)\(43\!\cdots\!60\)\( p^{20} T^{14} + \)\(10\!\cdots\!92\)\( p^{25} T^{15} + 989833625114591949 p^{30} T^{16} + 19754962156272 p^{35} T^{17} + 1471207574 p^{40} T^{18} + 18320 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
53 \( 1 - 4029665128 T^{2} + 7931455217506153830 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(98\!\cdots\!97\)\( T^{8} - \)\(75\!\cdots\!84\)\( T^{10} + \)\(48\!\cdots\!36\)\( T^{12} - \)\(27\!\cdots\!56\)\( T^{14} + \)\(13\!\cdots\!38\)\( T^{16} - \)\(64\!\cdots\!60\)\( T^{18} + \)\(27\!\cdots\!76\)\( T^{20} - \)\(64\!\cdots\!60\)\( p^{10} T^{22} + \)\(13\!\cdots\!38\)\( p^{20} T^{24} - \)\(27\!\cdots\!56\)\( p^{30} T^{26} + \)\(48\!\cdots\!36\)\( p^{40} T^{28} - \)\(75\!\cdots\!84\)\( p^{50} T^{30} + \)\(98\!\cdots\!97\)\( p^{60} T^{32} - \)\(10\!\cdots\!40\)\( p^{70} T^{34} + 7931455217506153830 p^{80} T^{36} - 4029665128 p^{90} T^{38} + p^{100} T^{40} \)
59 \( ( 1 - 31454 T + 4623295858 T^{2} - 124799446446410 T^{3} + 10014833728553042357 T^{4} - \)\(23\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!84\)\( T^{6} - \)\(29\!\cdots\!72\)\( T^{7} + \)\(14\!\cdots\!74\)\( T^{8} - \)\(26\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!16\)\( T^{10} - \)\(26\!\cdots\!00\)\( p^{5} T^{11} + \)\(14\!\cdots\!74\)\( p^{10} T^{12} - \)\(29\!\cdots\!72\)\( p^{15} T^{13} + \)\(13\!\cdots\!84\)\( p^{20} T^{14} - \)\(23\!\cdots\!44\)\( p^{25} T^{15} + 10014833728553042357 p^{30} T^{16} - 124799446446410 p^{35} T^{17} + 4623295858 p^{40} T^{18} - 31454 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
61 \( ( 1 - 36632 T + 5042515374 T^{2} - 191771260465784 T^{3} + 13110617571305596597 T^{4} - \)\(47\!\cdots\!08\)\( T^{5} + \)\(22\!\cdots\!44\)\( T^{6} - \)\(75\!\cdots\!28\)\( T^{7} + \)\(29\!\cdots\!38\)\( T^{8} - \)\(85\!\cdots\!84\)\( T^{9} + \)\(28\!\cdots\!44\)\( T^{10} - \)\(85\!\cdots\!84\)\( p^{5} T^{11} + \)\(29\!\cdots\!38\)\( p^{10} T^{12} - \)\(75\!\cdots\!28\)\( p^{15} T^{13} + \)\(22\!\cdots\!44\)\( p^{20} T^{14} - \)\(47\!\cdots\!08\)\( p^{25} T^{15} + 13110617571305596597 p^{30} T^{16} - 191771260465784 p^{35} T^{17} + 5042515374 p^{40} T^{18} - 36632 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
67 \( 1 - 10843590944 T^{2} + 59375557604770839670 T^{4} - \)\(22\!\cdots\!48\)\( T^{6} + \)\(66\!\cdots\!21\)\( T^{8} - \)\(16\!\cdots\!72\)\( T^{10} + \)\(35\!\cdots\!84\)\( T^{12} - \)\(67\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!50\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{18} + \)\(25\!\cdots\!48\)\( T^{20} - \)\(17\!\cdots\!24\)\( p^{10} T^{22} + \)\(11\!\cdots\!50\)\( p^{20} T^{24} - \)\(67\!\cdots\!28\)\( p^{30} T^{26} + \)\(35\!\cdots\!84\)\( p^{40} T^{28} - \)\(16\!\cdots\!72\)\( p^{50} T^{30} + \)\(66\!\cdots\!21\)\( p^{60} T^{32} - \)\(22\!\cdots\!48\)\( p^{70} T^{34} + 59375557604770839670 p^{80} T^{36} - 10843590944 p^{90} T^{38} + p^{100} T^{40} \)
71 \( ( 1 + 17444 T + 8202464246 T^{2} + 193853763963612 T^{3} + 37489454553586147677 T^{4} + \)\(91\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!88\)\( T^{6} + \)\(28\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!02\)\( T^{8} + \)\(65\!\cdots\!56\)\( T^{9} + \)\(60\!\cdots\!08\)\( T^{10} + \)\(65\!\cdots\!56\)\( p^{5} T^{11} + \)\(30\!\cdots\!02\)\( p^{10} T^{12} + \)\(28\!\cdots\!20\)\( p^{15} T^{13} + \)\(12\!\cdots\!88\)\( p^{20} T^{14} + \)\(91\!\cdots\!60\)\( p^{25} T^{15} + 37489454553586147677 p^{30} T^{16} + 193853763963612 p^{35} T^{17} + 8202464246 p^{40} T^{18} + 17444 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
73 \( ( 1 - 26284 T + 12244859470 T^{2} - 181128130380588 T^{3} + 73557964980644644029 T^{4} - \)\(63\!\cdots\!36\)\( T^{5} + \)\(30\!\cdots\!76\)\( T^{6} - \)\(17\!\cdots\!12\)\( T^{7} + \)\(92\!\cdots\!30\)\( T^{8} - \)\(38\!\cdots\!04\)\( T^{9} + \)\(21\!\cdots\!44\)\( T^{10} - \)\(38\!\cdots\!04\)\( p^{5} T^{11} + \)\(92\!\cdots\!30\)\( p^{10} T^{12} - \)\(17\!\cdots\!12\)\( p^{15} T^{13} + \)\(30\!\cdots\!76\)\( p^{20} T^{14} - \)\(63\!\cdots\!36\)\( p^{25} T^{15} + 73557964980644644029 p^{30} T^{16} - 181128130380588 p^{35} T^{17} + 12244859470 p^{40} T^{18} - 26284 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
79 \( 1 - 24163428736 T^{2} + \)\(30\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!72\)\( T^{6} + \)\(17\!\cdots\!37\)\( T^{8} - \)\(99\!\cdots\!00\)\( T^{10} + \)\(47\!\cdots\!76\)\( T^{12} - \)\(20\!\cdots\!04\)\( T^{14} + \)\(77\!\cdots\!26\)\( T^{16} - \)\(27\!\cdots\!44\)\( T^{18} + \)\(87\!\cdots\!76\)\( T^{20} - \)\(27\!\cdots\!44\)\( p^{10} T^{22} + \)\(77\!\cdots\!26\)\( p^{20} T^{24} - \)\(20\!\cdots\!04\)\( p^{30} T^{26} + \)\(47\!\cdots\!76\)\( p^{40} T^{28} - \)\(99\!\cdots\!00\)\( p^{50} T^{30} + \)\(17\!\cdots\!37\)\( p^{60} T^{32} - \)\(26\!\cdots\!72\)\( p^{70} T^{34} + \)\(30\!\cdots\!58\)\( p^{80} T^{36} - 24163428736 p^{90} T^{38} + p^{100} T^{40} \)
83 \( ( 1 + 112586 T + 19402311130 T^{2} + 1703864185686878 T^{3} + \)\(16\!\cdots\!21\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} + \)\(70\!\cdots\!32\)\( T^{7} + \)\(55\!\cdots\!30\)\( T^{8} + \)\(35\!\cdots\!76\)\( T^{9} + \)\(24\!\cdots\!96\)\( T^{10} + \)\(35\!\cdots\!76\)\( p^{5} T^{11} + \)\(55\!\cdots\!30\)\( p^{10} T^{12} + \)\(70\!\cdots\!32\)\( p^{15} T^{13} + \)\(10\!\cdots\!48\)\( p^{20} T^{14} + \)\(12\!\cdots\!00\)\( p^{25} T^{15} + \)\(16\!\cdots\!21\)\( p^{30} T^{16} + 1703864185686878 p^{35} T^{17} + 19402311130 p^{40} T^{18} + 112586 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
89 \( 1 - 53305266132 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{6} + \)\(27\!\cdots\!89\)\( T^{8} - \)\(27\!\cdots\!36\)\( T^{10} + \)\(22\!\cdots\!48\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(11\!\cdots\!22\)\( T^{16} - \)\(70\!\cdots\!80\)\( T^{18} + \)\(40\!\cdots\!20\)\( T^{20} - \)\(70\!\cdots\!80\)\( p^{10} T^{22} + \)\(11\!\cdots\!22\)\( p^{20} T^{24} - \)\(16\!\cdots\!56\)\( p^{30} T^{26} + \)\(22\!\cdots\!48\)\( p^{40} T^{28} - \)\(27\!\cdots\!36\)\( p^{50} T^{30} + \)\(27\!\cdots\!89\)\( p^{60} T^{32} - \)\(22\!\cdots\!96\)\( p^{70} T^{34} + \)\(13\!\cdots\!30\)\( p^{80} T^{36} - 53305266132 p^{90} T^{38} + p^{100} T^{40} \)
97 \( ( 1 - 3800 T + 22443771702 T^{2} - 608378862672152 T^{3} + \)\(32\!\cdots\!41\)\( T^{4} - \)\(82\!\cdots\!72\)\( T^{5} + \)\(43\!\cdots\!12\)\( T^{6} - \)\(80\!\cdots\!80\)\( T^{7} + \)\(47\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{9} + \)\(43\!\cdots\!80\)\( T^{10} - \)\(11\!\cdots\!16\)\( p^{5} T^{11} + \)\(47\!\cdots\!18\)\( p^{10} T^{12} - \)\(80\!\cdots\!80\)\( p^{15} T^{13} + \)\(43\!\cdots\!12\)\( p^{20} T^{14} - \)\(82\!\cdots\!72\)\( p^{25} T^{15} + \)\(32\!\cdots\!41\)\( p^{30} T^{16} - 608378862672152 p^{35} T^{17} + 22443771702 p^{40} T^{18} - 3800 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.81728614403723644388267869002, −1.69982054708962944453250210530, −1.45986035802684234415533365337, −1.41319706517908120612468827386, −1.34677983615161026632003164609, −1.23934492911654307476660130006, −1.23290262095998803697305439788, −1.19089852990158550651954350856, −1.15974720899120501801446705749, −1.14786310134413275425847705729, −1.06282133734592760273967624735, −1.05413936570198072968761225091, −1.04819452899975484955622706373, −1.01485808430919264534849035745, −1.00908497863368473188881856019, −0.994306549891834787683896177866, −0.838535927572167531915751277619, −0.60282495702453514548557770536, −0.47463076493746466079829611223, −0.44333883428693760051494763568, −0.34210417255504483212780509572, −0.14507698845203705786446552158, −0.05057070788975975033943595436, −0.04051073643350154233148873687, −0.03357237779973090259945177796, 0.03357237779973090259945177796, 0.04051073643350154233148873687, 0.05057070788975975033943595436, 0.14507698845203705786446552158, 0.34210417255504483212780509572, 0.44333883428693760051494763568, 0.47463076493746466079829611223, 0.60282495702453514548557770536, 0.838535927572167531915751277619, 0.994306549891834787683896177866, 1.00908497863368473188881856019, 1.01485808430919264534849035745, 1.04819452899975484955622706373, 1.05413936570198072968761225091, 1.06282133734592760273967624735, 1.14786310134413275425847705729, 1.15974720899120501801446705749, 1.19089852990158550651954350856, 1.23290262095998803697305439788, 1.23934492911654307476660130006, 1.34677983615161026632003164609, 1.41319706517908120612468827386, 1.45986035802684234415533365337, 1.69982054708962944453250210530, 1.81728614403723644388267869002

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

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