Properties

Label 32-531e16-1.1-c7e16-0-0
Degree $32$
Conductor $3.995\times 10^{43}$
Sign $1$
Analytic cond. $3.28515\times 10^{35}$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $16$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 519·4-s + 68·5-s − 2.34e3·7-s − 3.71e3·8-s + 408·10-s − 898·11-s − 8.17e3·13-s − 1.40e4·14-s + 1.01e5·16-s + 4.49e4·17-s − 4.01e4·19-s − 3.52e4·20-s − 5.38e3·22-s + 2.83e3·23-s − 4.79e5·25-s − 4.90e4·26-s + 1.21e6·28-s − 1.44e5·29-s − 1.41e5·31-s + 9.66e5·32-s + 2.69e5·34-s − 1.59e5·35-s − 2.97e5·37-s − 2.40e5·38-s − 2.52e5·40-s − 6.59e5·41-s + ⋯
L(s)  = 1  + 0.530·2-s − 4.05·4-s + 0.243·5-s − 2.58·7-s − 2.56·8-s + 0.129·10-s − 0.203·11-s − 1.03·13-s − 1.36·14-s + 6.16·16-s + 2.22·17-s − 1.34·19-s − 0.986·20-s − 0.107·22-s + 0.0485·23-s − 6.14·25-s − 0.547·26-s + 10.4·28-s − 1.09·29-s − 0.854·31-s + 5.21·32-s + 1.17·34-s − 0.628·35-s − 0.967·37-s − 0.711·38-s − 0.624·40-s − 1.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 59^{16}\)
Sign: $1$
Analytic conductor: \(3.28515\times 10^{35}\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{531} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(16\)
Selberg data: \((32,\ 3^{32} \cdot 59^{16} ,\ ( \ : [7/2]^{16} ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( ( 1 + p^{3} T )^{16} \)
good2 \( 1 - 3 p T + 555 T^{2} - 2729 T^{3} + 90557 p T^{4} - 801421 T^{5} + 23563035 p T^{6} - 6064111 p^{5} T^{7} + 319507921 p^{5} T^{8} - 1249075169 p^{5} T^{9} + 30022780525 p^{6} T^{10} - 28089249987 p^{8} T^{11} + 78484723671 p^{12} T^{12} - 4464698946637 p^{8} T^{13} + 93818649401511 p^{9} T^{14} - 39580945805855 p^{12} T^{15} + 788182218735117 p^{13} T^{16} - 39580945805855 p^{19} T^{17} + 93818649401511 p^{23} T^{18} - 4464698946637 p^{29} T^{19} + 78484723671 p^{40} T^{20} - 28089249987 p^{43} T^{21} + 30022780525 p^{48} T^{22} - 1249075169 p^{54} T^{23} + 319507921 p^{61} T^{24} - 6064111 p^{68} T^{25} + 23563035 p^{71} T^{26} - 801421 p^{77} T^{27} + 90557 p^{85} T^{28} - 2729 p^{91} T^{29} + 555 p^{98} T^{30} - 3 p^{106} T^{31} + p^{112} T^{32} \)
5 \( 1 - 68 T + 484439 T^{2} - 32331978 T^{3} + 23151305886 p T^{4} - 1553151288618 p T^{5} + 3706537404251679 p T^{6} - 59466026220969652 p^{2} T^{7} + \)\(23\!\cdots\!29\)\( T^{8} - \)\(51\!\cdots\!44\)\( p T^{9} + \)\(19\!\cdots\!12\)\( p^{3} T^{10} - \)\(29\!\cdots\!64\)\( p^{3} T^{11} + \)\(36\!\cdots\!38\)\( p^{4} T^{12} - \)\(26\!\cdots\!52\)\( p^{6} T^{13} + \)\(24\!\cdots\!54\)\( p^{7} T^{14} - \)\(49\!\cdots\!28\)\( p^{7} T^{15} + \)\(39\!\cdots\!32\)\( p^{8} T^{16} - \)\(49\!\cdots\!28\)\( p^{14} T^{17} + \)\(24\!\cdots\!54\)\( p^{21} T^{18} - \)\(26\!\cdots\!52\)\( p^{27} T^{19} + \)\(36\!\cdots\!38\)\( p^{32} T^{20} - \)\(29\!\cdots\!64\)\( p^{38} T^{21} + \)\(19\!\cdots\!12\)\( p^{45} T^{22} - \)\(51\!\cdots\!44\)\( p^{50} T^{23} + \)\(23\!\cdots\!29\)\( p^{56} T^{24} - 59466026220969652 p^{65} T^{25} + 3706537404251679 p^{71} T^{26} - 1553151288618 p^{78} T^{27} + 23151305886 p^{85} T^{28} - 32331978 p^{91} T^{29} + 484439 p^{98} T^{30} - 68 p^{105} T^{31} + p^{112} T^{32} \)
7 \( 1 + 2343 T + 8386396 T^{2} + 340924638 p^{2} T^{3} + 36305360326168 T^{4} + 61555776166460035 T^{5} + \)\(10\!\cdots\!22\)\( T^{6} + \)\(15\!\cdots\!89\)\( T^{7} + \)\(22\!\cdots\!20\)\( T^{8} + \)\(28\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!94\)\( p T^{10} + \)\(42\!\cdots\!75\)\( T^{11} + \)\(48\!\cdots\!01\)\( T^{12} + \)\(51\!\cdots\!20\)\( T^{13} + \)\(52\!\cdots\!84\)\( T^{14} + \)\(72\!\cdots\!50\)\( p T^{15} + \)\(47\!\cdots\!48\)\( T^{16} + \)\(72\!\cdots\!50\)\( p^{8} T^{17} + \)\(52\!\cdots\!84\)\( p^{14} T^{18} + \)\(51\!\cdots\!20\)\( p^{21} T^{19} + \)\(48\!\cdots\!01\)\( p^{28} T^{20} + \)\(42\!\cdots\!75\)\( p^{35} T^{21} + \)\(52\!\cdots\!94\)\( p^{43} T^{22} + \)\(28\!\cdots\!54\)\( p^{49} T^{23} + \)\(22\!\cdots\!20\)\( p^{56} T^{24} + \)\(15\!\cdots\!89\)\( p^{63} T^{25} + \)\(10\!\cdots\!22\)\( p^{70} T^{26} + 61555776166460035 p^{77} T^{27} + 36305360326168 p^{84} T^{28} + 340924638 p^{93} T^{29} + 8386396 p^{98} T^{30} + 2343 p^{105} T^{31} + p^{112} T^{32} \)
11 \( 1 + 898 T + 163741540 T^{2} + 42234715948 T^{3} + 12957347483852297 T^{4} - 3797236701767068896 T^{5} + \)\(66\!\cdots\!08\)\( T^{6} - \)\(48\!\cdots\!06\)\( T^{7} + \)\(25\!\cdots\!97\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{9} + \)\(77\!\cdots\!64\)\( T^{10} - \)\(84\!\cdots\!72\)\( T^{11} + \)\(20\!\cdots\!06\)\( T^{12} - \)\(20\!\cdots\!12\)\( T^{13} + \)\(46\!\cdots\!24\)\( T^{14} - \)\(41\!\cdots\!24\)\( T^{15} + \)\(94\!\cdots\!38\)\( T^{16} - \)\(41\!\cdots\!24\)\( p^{7} T^{17} + \)\(46\!\cdots\!24\)\( p^{14} T^{18} - \)\(20\!\cdots\!12\)\( p^{21} T^{19} + \)\(20\!\cdots\!06\)\( p^{28} T^{20} - \)\(84\!\cdots\!72\)\( p^{35} T^{21} + \)\(77\!\cdots\!64\)\( p^{42} T^{22} - \)\(25\!\cdots\!00\)\( p^{49} T^{23} + \)\(25\!\cdots\!97\)\( p^{56} T^{24} - \)\(48\!\cdots\!06\)\( p^{63} T^{25} + \)\(66\!\cdots\!08\)\( p^{70} T^{26} - 3797236701767068896 p^{77} T^{27} + 12957347483852297 p^{84} T^{28} + 42234715948 p^{91} T^{29} + 163741540 p^{98} T^{30} + 898 p^{105} T^{31} + p^{112} T^{32} \)
13 \( 1 + 8172 T + 511935818 T^{2} + 3986968146014 T^{3} + 135894539819802669 T^{4} + 79319665280264615160 p T^{5} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!42\)\( T^{7} + \)\(26\!\cdots\!33\)\( p T^{8} + \)\(19\!\cdots\!66\)\( p T^{9} + \)\(38\!\cdots\!62\)\( T^{10} + \)\(26\!\cdots\!94\)\( T^{11} + \)\(35\!\cdots\!34\)\( T^{12} + \)\(24\!\cdots\!14\)\( T^{13} + \)\(28\!\cdots\!94\)\( T^{14} + \)\(17\!\cdots\!86\)\( T^{15} + \)\(19\!\cdots\!62\)\( T^{16} + \)\(17\!\cdots\!86\)\( p^{7} T^{17} + \)\(28\!\cdots\!94\)\( p^{14} T^{18} + \)\(24\!\cdots\!14\)\( p^{21} T^{19} + \)\(35\!\cdots\!34\)\( p^{28} T^{20} + \)\(26\!\cdots\!94\)\( p^{35} T^{21} + \)\(38\!\cdots\!62\)\( p^{42} T^{22} + \)\(19\!\cdots\!66\)\( p^{50} T^{23} + \)\(26\!\cdots\!33\)\( p^{57} T^{24} + \)\(18\!\cdots\!42\)\( p^{63} T^{25} + \)\(24\!\cdots\!00\)\( p^{70} T^{26} + 79319665280264615160 p^{78} T^{27} + 135894539819802669 p^{84} T^{28} + 3986968146014 p^{91} T^{29} + 511935818 p^{98} T^{30} + 8172 p^{105} T^{31} + p^{112} T^{32} \)
17 \( 1 - 44985 T + 5494681630 T^{2} - 205790956761078 T^{3} + 14099821456028369852 T^{4} - \)\(45\!\cdots\!21\)\( T^{5} + \)\(22\!\cdots\!54\)\( T^{6} - \)\(38\!\cdots\!13\)\( p T^{7} + \)\(26\!\cdots\!36\)\( T^{8} - \)\(66\!\cdots\!74\)\( T^{9} + \)\(22\!\cdots\!92\)\( T^{10} - \)\(52\!\cdots\!47\)\( T^{11} + \)\(15\!\cdots\!65\)\( T^{12} - \)\(32\!\cdots\!92\)\( T^{13} + \)\(87\!\cdots\!76\)\( T^{14} - \)\(16\!\cdots\!14\)\( T^{15} + \)\(39\!\cdots\!08\)\( T^{16} - \)\(16\!\cdots\!14\)\( p^{7} T^{17} + \)\(87\!\cdots\!76\)\( p^{14} T^{18} - \)\(32\!\cdots\!92\)\( p^{21} T^{19} + \)\(15\!\cdots\!65\)\( p^{28} T^{20} - \)\(52\!\cdots\!47\)\( p^{35} T^{21} + \)\(22\!\cdots\!92\)\( p^{42} T^{22} - \)\(66\!\cdots\!74\)\( p^{49} T^{23} + \)\(26\!\cdots\!36\)\( p^{56} T^{24} - \)\(38\!\cdots\!13\)\( p^{64} T^{25} + \)\(22\!\cdots\!54\)\( p^{70} T^{26} - \)\(45\!\cdots\!21\)\( p^{77} T^{27} + 14099821456028369852 p^{84} T^{28} - 205790956761078 p^{91} T^{29} + 5494681630 p^{98} T^{30} - 44985 p^{105} T^{31} + p^{112} T^{32} \)
19 \( 1 + 40137 T + 6809402349 T^{2} + 225411665068823 T^{3} + 21791992588224206783 T^{4} + \)\(60\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!10\)\( p T^{6} + \)\(10\!\cdots\!70\)\( T^{7} + \)\(67\!\cdots\!63\)\( T^{8} + \)\(13\!\cdots\!21\)\( T^{9} + \)\(86\!\cdots\!99\)\( T^{10} + \)\(16\!\cdots\!81\)\( T^{11} + \)\(10\!\cdots\!91\)\( T^{12} + \)\(18\!\cdots\!18\)\( T^{13} + \)\(10\!\cdots\!70\)\( T^{14} + \)\(18\!\cdots\!20\)\( T^{15} + \)\(10\!\cdots\!88\)\( T^{16} + \)\(18\!\cdots\!20\)\( p^{7} T^{17} + \)\(10\!\cdots\!70\)\( p^{14} T^{18} + \)\(18\!\cdots\!18\)\( p^{21} T^{19} + \)\(10\!\cdots\!91\)\( p^{28} T^{20} + \)\(16\!\cdots\!81\)\( p^{35} T^{21} + \)\(86\!\cdots\!99\)\( p^{42} T^{22} + \)\(13\!\cdots\!21\)\( p^{49} T^{23} + \)\(67\!\cdots\!63\)\( p^{56} T^{24} + \)\(10\!\cdots\!70\)\( p^{63} T^{25} + \)\(23\!\cdots\!10\)\( p^{71} T^{26} + \)\(60\!\cdots\!90\)\( p^{77} T^{27} + 21791992588224206783 p^{84} T^{28} + 225411665068823 p^{91} T^{29} + 6809402349 p^{98} T^{30} + 40137 p^{105} T^{31} + p^{112} T^{32} \)
23 \( 1 - 2833 T + 20814787185 T^{2} + 648124018861281 T^{3} + \)\(21\!\cdots\!99\)\( T^{4} + \)\(12\!\cdots\!26\)\( T^{5} + \)\(17\!\cdots\!58\)\( T^{6} + \)\(12\!\cdots\!02\)\( T^{7} + \)\(12\!\cdots\!27\)\( T^{8} + \)\(92\!\cdots\!11\)\( T^{9} + \)\(72\!\cdots\!23\)\( T^{10} + \)\(52\!\cdots\!11\)\( T^{11} + \)\(15\!\cdots\!25\)\( p T^{12} + \)\(24\!\cdots\!90\)\( T^{13} + \)\(15\!\cdots\!54\)\( T^{14} + \)\(97\!\cdots\!88\)\( T^{15} + \)\(56\!\cdots\!48\)\( T^{16} + \)\(97\!\cdots\!88\)\( p^{7} T^{17} + \)\(15\!\cdots\!54\)\( p^{14} T^{18} + \)\(24\!\cdots\!90\)\( p^{21} T^{19} + \)\(15\!\cdots\!25\)\( p^{29} T^{20} + \)\(52\!\cdots\!11\)\( p^{35} T^{21} + \)\(72\!\cdots\!23\)\( p^{42} T^{22} + \)\(92\!\cdots\!11\)\( p^{49} T^{23} + \)\(12\!\cdots\!27\)\( p^{56} T^{24} + \)\(12\!\cdots\!02\)\( p^{63} T^{25} + \)\(17\!\cdots\!58\)\( p^{70} T^{26} + \)\(12\!\cdots\!26\)\( p^{77} T^{27} + \)\(21\!\cdots\!99\)\( p^{84} T^{28} + 648124018861281 p^{91} T^{29} + 20814787185 p^{98} T^{30} - 2833 p^{105} T^{31} + p^{112} T^{32} \)
29 \( 1 + 144375 T + 144322094863 T^{2} + 16887377443378959 T^{3} + \)\(99\!\cdots\!65\)\( T^{4} + \)\(10\!\cdots\!38\)\( T^{5} + \)\(44\!\cdots\!34\)\( T^{6} + \)\(43\!\cdots\!22\)\( T^{7} + \)\(15\!\cdots\!29\)\( T^{8} + \)\(14\!\cdots\!47\)\( T^{9} + \)\(40\!\cdots\!45\)\( T^{10} + \)\(38\!\cdots\!37\)\( T^{11} + \)\(91\!\cdots\!07\)\( T^{12} + \)\(89\!\cdots\!82\)\( T^{13} + \)\(18\!\cdots\!54\)\( T^{14} + \)\(17\!\cdots\!60\)\( T^{15} + \)\(32\!\cdots\!44\)\( T^{16} + \)\(17\!\cdots\!60\)\( p^{7} T^{17} + \)\(18\!\cdots\!54\)\( p^{14} T^{18} + \)\(89\!\cdots\!82\)\( p^{21} T^{19} + \)\(91\!\cdots\!07\)\( p^{28} T^{20} + \)\(38\!\cdots\!37\)\( p^{35} T^{21} + \)\(40\!\cdots\!45\)\( p^{42} T^{22} + \)\(14\!\cdots\!47\)\( p^{49} T^{23} + \)\(15\!\cdots\!29\)\( p^{56} T^{24} + \)\(43\!\cdots\!22\)\( p^{63} T^{25} + \)\(44\!\cdots\!34\)\( p^{70} T^{26} + \)\(10\!\cdots\!38\)\( p^{77} T^{27} + \)\(99\!\cdots\!65\)\( p^{84} T^{28} + 16887377443378959 p^{91} T^{29} + 144322094863 p^{98} T^{30} + 144375 p^{105} T^{31} + p^{112} T^{32} \)
31 \( 1 + 141759 T + 311746829431 T^{2} + 38304726706398301 T^{3} + \)\(14\!\cdots\!43\)\( p T^{4} + \)\(48\!\cdots\!90\)\( T^{5} + \)\(44\!\cdots\!86\)\( T^{6} + \)\(38\!\cdots\!98\)\( T^{7} + \)\(29\!\cdots\!53\)\( T^{8} + \)\(21\!\cdots\!79\)\( T^{9} + \)\(15\!\cdots\!33\)\( T^{10} + \)\(86\!\cdots\!39\)\( T^{11} + \)\(64\!\cdots\!39\)\( T^{12} + \)\(28\!\cdots\!66\)\( T^{13} + \)\(22\!\cdots\!42\)\( T^{14} + \)\(82\!\cdots\!24\)\( T^{15} + \)\(66\!\cdots\!32\)\( T^{16} + \)\(82\!\cdots\!24\)\( p^{7} T^{17} + \)\(22\!\cdots\!42\)\( p^{14} T^{18} + \)\(28\!\cdots\!66\)\( p^{21} T^{19} + \)\(64\!\cdots\!39\)\( p^{28} T^{20} + \)\(86\!\cdots\!39\)\( p^{35} T^{21} + \)\(15\!\cdots\!33\)\( p^{42} T^{22} + \)\(21\!\cdots\!79\)\( p^{49} T^{23} + \)\(29\!\cdots\!53\)\( p^{56} T^{24} + \)\(38\!\cdots\!98\)\( p^{63} T^{25} + \)\(44\!\cdots\!86\)\( p^{70} T^{26} + \)\(48\!\cdots\!90\)\( p^{77} T^{27} + \)\(14\!\cdots\!43\)\( p^{85} T^{28} + 38304726706398301 p^{91} T^{29} + 311746829431 p^{98} T^{30} + 141759 p^{105} T^{31} + p^{112} T^{32} \)
37 \( 1 + 297971 T + 17316545788 p T^{2} + 174031755215644656 T^{3} + \)\(22\!\cdots\!82\)\( T^{4} + \)\(59\!\cdots\!53\)\( T^{5} + \)\(56\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!09\)\( T^{7} + \)\(11\!\cdots\!34\)\( T^{8} + \)\(27\!\cdots\!96\)\( T^{9} + \)\(18\!\cdots\!42\)\( T^{10} + \)\(42\!\cdots\!61\)\( T^{11} + \)\(24\!\cdots\!29\)\( T^{12} + \)\(55\!\cdots\!08\)\( T^{13} + \)\(29\!\cdots\!76\)\( T^{14} + \)\(61\!\cdots\!82\)\( T^{15} + \)\(29\!\cdots\!52\)\( T^{16} + \)\(61\!\cdots\!82\)\( p^{7} T^{17} + \)\(29\!\cdots\!76\)\( p^{14} T^{18} + \)\(55\!\cdots\!08\)\( p^{21} T^{19} + \)\(24\!\cdots\!29\)\( p^{28} T^{20} + \)\(42\!\cdots\!61\)\( p^{35} T^{21} + \)\(18\!\cdots\!42\)\( p^{42} T^{22} + \)\(27\!\cdots\!96\)\( p^{49} T^{23} + \)\(11\!\cdots\!34\)\( p^{56} T^{24} + \)\(14\!\cdots\!09\)\( p^{63} T^{25} + \)\(56\!\cdots\!16\)\( p^{70} T^{26} + \)\(59\!\cdots\!53\)\( p^{77} T^{27} + \)\(22\!\cdots\!82\)\( p^{84} T^{28} + 174031755215644656 p^{91} T^{29} + 17316545788 p^{99} T^{30} + 297971 p^{105} T^{31} + p^{112} T^{32} \)
41 \( 1 + 659077 T + 1837498595458 T^{2} + 1066858548141989670 T^{3} + \)\(16\!\cdots\!60\)\( T^{4} + \)\(89\!\cdots\!05\)\( T^{5} + \)\(10\!\cdots\!34\)\( T^{6} + \)\(50\!\cdots\!97\)\( T^{7} + \)\(46\!\cdots\!44\)\( T^{8} + \)\(21\!\cdots\!50\)\( T^{9} + \)\(17\!\cdots\!84\)\( T^{10} + \)\(74\!\cdots\!71\)\( T^{11} + \)\(51\!\cdots\!81\)\( T^{12} + \)\(20\!\cdots\!84\)\( T^{13} + \)\(12\!\cdots\!48\)\( T^{14} + \)\(48\!\cdots\!94\)\( T^{15} + \)\(27\!\cdots\!76\)\( T^{16} + \)\(48\!\cdots\!94\)\( p^{7} T^{17} + \)\(12\!\cdots\!48\)\( p^{14} T^{18} + \)\(20\!\cdots\!84\)\( p^{21} T^{19} + \)\(51\!\cdots\!81\)\( p^{28} T^{20} + \)\(74\!\cdots\!71\)\( p^{35} T^{21} + \)\(17\!\cdots\!84\)\( p^{42} T^{22} + \)\(21\!\cdots\!50\)\( p^{49} T^{23} + \)\(46\!\cdots\!44\)\( p^{56} T^{24} + \)\(50\!\cdots\!97\)\( p^{63} T^{25} + \)\(10\!\cdots\!34\)\( p^{70} T^{26} + \)\(89\!\cdots\!05\)\( p^{77} T^{27} + \)\(16\!\cdots\!60\)\( p^{84} T^{28} + 1066858548141989670 p^{91} T^{29} + 1837498595458 p^{98} T^{30} + 659077 p^{105} T^{31} + p^{112} T^{32} \)
43 \( 1 + 1431608 T + 2523178123406 T^{2} + 2698404381912396320 T^{3} + \)\(31\!\cdots\!79\)\( T^{4} + \)\(28\!\cdots\!56\)\( T^{5} + \)\(26\!\cdots\!02\)\( T^{6} + \)\(20\!\cdots\!36\)\( T^{7} + \)\(16\!\cdots\!25\)\( T^{8} + \)\(11\!\cdots\!96\)\( T^{9} + \)\(82\!\cdots\!60\)\( T^{10} + \)\(53\!\cdots\!04\)\( T^{11} + \)\(34\!\cdots\!10\)\( T^{12} + \)\(20\!\cdots\!72\)\( T^{13} + \)\(11\!\cdots\!72\)\( T^{14} + \)\(65\!\cdots\!24\)\( T^{15} + \)\(35\!\cdots\!30\)\( T^{16} + \)\(65\!\cdots\!24\)\( p^{7} T^{17} + \)\(11\!\cdots\!72\)\( p^{14} T^{18} + \)\(20\!\cdots\!72\)\( p^{21} T^{19} + \)\(34\!\cdots\!10\)\( p^{28} T^{20} + \)\(53\!\cdots\!04\)\( p^{35} T^{21} + \)\(82\!\cdots\!60\)\( p^{42} T^{22} + \)\(11\!\cdots\!96\)\( p^{49} T^{23} + \)\(16\!\cdots\!25\)\( p^{56} T^{24} + \)\(20\!\cdots\!36\)\( p^{63} T^{25} + \)\(26\!\cdots\!02\)\( p^{70} T^{26} + \)\(28\!\cdots\!56\)\( p^{77} T^{27} + \)\(31\!\cdots\!79\)\( p^{84} T^{28} + 2698404381912396320 p^{91} T^{29} + 2523178123406 p^{98} T^{30} + 1431608 p^{105} T^{31} + p^{112} T^{32} \)
47 \( 1 - 1574073 T + 3535103246343 T^{2} - 4432622288871442935 T^{3} + \)\(66\!\cdots\!87\)\( T^{4} - \)\(72\!\cdots\!62\)\( T^{5} + \)\(87\!\cdots\!78\)\( T^{6} - \)\(85\!\cdots\!10\)\( T^{7} + \)\(88\!\cdots\!95\)\( T^{8} - \)\(79\!\cdots\!05\)\( T^{9} + \)\(73\!\cdots\!73\)\( T^{10} - \)\(60\!\cdots\!01\)\( T^{11} + \)\(51\!\cdots\!71\)\( T^{12} - \)\(39\!\cdots\!70\)\( T^{13} + \)\(31\!\cdots\!06\)\( T^{14} - \)\(22\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!68\)\( T^{16} - \)\(22\!\cdots\!08\)\( p^{7} T^{17} + \)\(31\!\cdots\!06\)\( p^{14} T^{18} - \)\(39\!\cdots\!70\)\( p^{21} T^{19} + \)\(51\!\cdots\!71\)\( p^{28} T^{20} - \)\(60\!\cdots\!01\)\( p^{35} T^{21} + \)\(73\!\cdots\!73\)\( p^{42} T^{22} - \)\(79\!\cdots\!05\)\( p^{49} T^{23} + \)\(88\!\cdots\!95\)\( p^{56} T^{24} - \)\(85\!\cdots\!10\)\( p^{63} T^{25} + \)\(87\!\cdots\!78\)\( p^{70} T^{26} - \)\(72\!\cdots\!62\)\( p^{77} T^{27} + \)\(66\!\cdots\!87\)\( p^{84} T^{28} - 4432622288871442935 p^{91} T^{29} + 3535103246343 p^{98} T^{30} - 1574073 p^{105} T^{31} + p^{112} T^{32} \)
53 \( 1 + 587736 T + 10691953380791 T^{2} + 4079852343577778034 T^{3} + \)\(55\!\cdots\!98\)\( T^{4} + \)\(21\!\cdots\!46\)\( p T^{5} + \)\(19\!\cdots\!63\)\( T^{6} + \)\(78\!\cdots\!56\)\( T^{7} + \)\(48\!\cdots\!85\)\( T^{8} - \)\(45\!\cdots\!76\)\( T^{9} + \)\(99\!\cdots\!36\)\( T^{10} - \)\(19\!\cdots\!56\)\( T^{11} + \)\(17\!\cdots\!18\)\( T^{12} - \)\(43\!\cdots\!68\)\( T^{13} + \)\(46\!\cdots\!02\)\( p T^{14} - \)\(68\!\cdots\!84\)\( T^{15} + \)\(31\!\cdots\!36\)\( T^{16} - \)\(68\!\cdots\!84\)\( p^{7} T^{17} + \)\(46\!\cdots\!02\)\( p^{15} T^{18} - \)\(43\!\cdots\!68\)\( p^{21} T^{19} + \)\(17\!\cdots\!18\)\( p^{28} T^{20} - \)\(19\!\cdots\!56\)\( p^{35} T^{21} + \)\(99\!\cdots\!36\)\( p^{42} T^{22} - \)\(45\!\cdots\!76\)\( p^{49} T^{23} + \)\(48\!\cdots\!85\)\( p^{56} T^{24} + \)\(78\!\cdots\!56\)\( p^{63} T^{25} + \)\(19\!\cdots\!63\)\( p^{70} T^{26} + \)\(21\!\cdots\!46\)\( p^{78} T^{27} + \)\(55\!\cdots\!98\)\( p^{84} T^{28} + 4079852343577778034 p^{91} T^{29} + 10691953380791 p^{98} T^{30} + 587736 p^{105} T^{31} + p^{112} T^{32} \)
61 \( 1 + 6117131 T + 47369369697139 T^{2} + \)\(20\!\cdots\!75\)\( T^{3} + \)\(95\!\cdots\!09\)\( T^{4} + \)\(33\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!62\)\( T^{6} + \)\(34\!\cdots\!50\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} + \)\(25\!\cdots\!15\)\( T^{9} + \)\(65\!\cdots\!69\)\( T^{10} + \)\(14\!\cdots\!65\)\( T^{11} + \)\(33\!\cdots\!91\)\( T^{12} + \)\(68\!\cdots\!26\)\( T^{13} + \)\(13\!\cdots\!10\)\( T^{14} + \)\(25\!\cdots\!80\)\( T^{15} + \)\(48\!\cdots\!56\)\( T^{16} + \)\(25\!\cdots\!80\)\( p^{7} T^{17} + \)\(13\!\cdots\!10\)\( p^{14} T^{18} + \)\(68\!\cdots\!26\)\( p^{21} T^{19} + \)\(33\!\cdots\!91\)\( p^{28} T^{20} + \)\(14\!\cdots\!65\)\( p^{35} T^{21} + \)\(65\!\cdots\!69\)\( p^{42} T^{22} + \)\(25\!\cdots\!15\)\( p^{49} T^{23} + \)\(10\!\cdots\!13\)\( p^{56} T^{24} + \)\(34\!\cdots\!50\)\( p^{63} T^{25} + \)\(11\!\cdots\!62\)\( p^{70} T^{26} + \)\(33\!\cdots\!50\)\( p^{77} T^{27} + \)\(95\!\cdots\!09\)\( p^{84} T^{28} + \)\(20\!\cdots\!75\)\( p^{91} T^{29} + 47369369697139 p^{98} T^{30} + 6117131 p^{105} T^{31} + p^{112} T^{32} \)
67 \( 1 + 16518710 T + 196069542561581 T^{2} + \)\(16\!\cdots\!34\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(74\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!41\)\( T^{6} + \)\(19\!\cdots\!50\)\( T^{7} + \)\(86\!\cdots\!89\)\( T^{8} + \)\(34\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!52\)\( T^{10} + \)\(45\!\cdots\!88\)\( T^{11} + \)\(14\!\cdots\!34\)\( T^{12} + \)\(44\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!42\)\( T^{14} + \)\(34\!\cdots\!44\)\( T^{15} + \)\(87\!\cdots\!28\)\( T^{16} + \)\(34\!\cdots\!44\)\( p^{7} T^{17} + \)\(12\!\cdots\!42\)\( p^{14} T^{18} + \)\(44\!\cdots\!56\)\( p^{21} T^{19} + \)\(14\!\cdots\!34\)\( p^{28} T^{20} + \)\(45\!\cdots\!88\)\( p^{35} T^{21} + \)\(13\!\cdots\!52\)\( p^{42} T^{22} + \)\(34\!\cdots\!08\)\( p^{49} T^{23} + \)\(86\!\cdots\!89\)\( p^{56} T^{24} + \)\(19\!\cdots\!50\)\( p^{63} T^{25} + \)\(40\!\cdots\!41\)\( p^{70} T^{26} + \)\(74\!\cdots\!54\)\( p^{77} T^{27} + \)\(12\!\cdots\!16\)\( p^{84} T^{28} + \)\(16\!\cdots\!34\)\( p^{91} T^{29} + 196069542561581 p^{98} T^{30} + 16518710 p^{105} T^{31} + p^{112} T^{32} \)
71 \( 1 - 10882582 T + 86794164883412 T^{2} - \)\(58\!\cdots\!10\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} - \)\(19\!\cdots\!48\)\( T^{5} + \)\(96\!\cdots\!06\)\( T^{6} - \)\(45\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} - \)\(83\!\cdots\!06\)\( T^{9} + \)\(33\!\cdots\!62\)\( T^{10} - \)\(12\!\cdots\!22\)\( T^{11} + \)\(64\!\cdots\!90\)\( p T^{12} - \)\(15\!\cdots\!30\)\( T^{13} + \)\(52\!\cdots\!22\)\( T^{14} - \)\(16\!\cdots\!10\)\( T^{15} + \)\(52\!\cdots\!10\)\( T^{16} - \)\(16\!\cdots\!10\)\( p^{7} T^{17} + \)\(52\!\cdots\!22\)\( p^{14} T^{18} - \)\(15\!\cdots\!30\)\( p^{21} T^{19} + \)\(64\!\cdots\!90\)\( p^{29} T^{20} - \)\(12\!\cdots\!22\)\( p^{35} T^{21} + \)\(33\!\cdots\!62\)\( p^{42} T^{22} - \)\(83\!\cdots\!06\)\( p^{49} T^{23} + \)\(19\!\cdots\!21\)\( p^{56} T^{24} - \)\(45\!\cdots\!92\)\( p^{63} T^{25} + \)\(96\!\cdots\!06\)\( p^{70} T^{26} - \)\(19\!\cdots\!48\)\( p^{77} T^{27} + \)\(35\!\cdots\!71\)\( p^{84} T^{28} - \)\(58\!\cdots\!10\)\( p^{91} T^{29} + 86794164883412 p^{98} T^{30} - 10882582 p^{105} T^{31} + p^{112} T^{32} \)
73 \( 1 + 21097441 T + 337490184288491 T^{2} + \)\(38\!\cdots\!29\)\( T^{3} + \)\(38\!\cdots\!67\)\( T^{4} + \)\(31\!\cdots\!82\)\( T^{5} + \)\(23\!\cdots\!22\)\( T^{6} + \)\(15\!\cdots\!58\)\( T^{7} + \)\(93\!\cdots\!75\)\( T^{8} + \)\(51\!\cdots\!81\)\( T^{9} + \)\(26\!\cdots\!81\)\( T^{10} + \)\(12\!\cdots\!83\)\( T^{11} + \)\(55\!\cdots\!07\)\( T^{12} + \)\(22\!\cdots\!10\)\( T^{13} + \)\(89\!\cdots\!66\)\( T^{14} + \)\(32\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!20\)\( T^{16} + \)\(32\!\cdots\!00\)\( p^{7} T^{17} + \)\(89\!\cdots\!66\)\( p^{14} T^{18} + \)\(22\!\cdots\!10\)\( p^{21} T^{19} + \)\(55\!\cdots\!07\)\( p^{28} T^{20} + \)\(12\!\cdots\!83\)\( p^{35} T^{21} + \)\(26\!\cdots\!81\)\( p^{42} T^{22} + \)\(51\!\cdots\!81\)\( p^{49} T^{23} + \)\(93\!\cdots\!75\)\( p^{56} T^{24} + \)\(15\!\cdots\!58\)\( p^{63} T^{25} + \)\(23\!\cdots\!22\)\( p^{70} T^{26} + \)\(31\!\cdots\!82\)\( p^{77} T^{27} + \)\(38\!\cdots\!67\)\( p^{84} T^{28} + \)\(38\!\cdots\!29\)\( p^{91} T^{29} + 337490184288491 p^{98} T^{30} + 21097441 p^{105} T^{31} + p^{112} T^{32} \)
79 \( 1 + 3784458 T + 142950984155520 T^{2} + \)\(56\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!77\)\( T^{4} + \)\(45\!\cdots\!92\)\( T^{5} + \)\(55\!\cdots\!72\)\( T^{6} + \)\(24\!\cdots\!74\)\( T^{7} + \)\(22\!\cdots\!77\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{9} + \)\(74\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!66\)\( T^{12} + \)\(89\!\cdots\!28\)\( T^{13} + \)\(50\!\cdots\!60\)\( T^{14} + \)\(20\!\cdots\!72\)\( T^{15} + \)\(10\!\cdots\!78\)\( T^{16} + \)\(20\!\cdots\!72\)\( p^{7} T^{17} + \)\(50\!\cdots\!60\)\( p^{14} T^{18} + \)\(89\!\cdots\!28\)\( p^{21} T^{19} + \)\(20\!\cdots\!66\)\( p^{28} T^{20} + \)\(33\!\cdots\!40\)\( p^{35} T^{21} + \)\(74\!\cdots\!80\)\( p^{42} T^{22} + \)\(10\!\cdots\!96\)\( p^{49} T^{23} + \)\(22\!\cdots\!77\)\( p^{56} T^{24} + \)\(24\!\cdots\!74\)\( p^{63} T^{25} + \)\(55\!\cdots\!72\)\( p^{70} T^{26} + \)\(45\!\cdots\!92\)\( p^{77} T^{27} + \)\(10\!\cdots\!77\)\( p^{84} T^{28} + \)\(56\!\cdots\!12\)\( p^{91} T^{29} + 142950984155520 p^{98} T^{30} + 3784458 p^{105} T^{31} + p^{112} T^{32} \)
83 \( 1 - 1951425 T + 177736299568080 T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!06\)\( T^{4} + \)\(59\!\cdots\!05\)\( T^{5} + \)\(10\!\cdots\!22\)\( T^{6} + \)\(11\!\cdots\!03\)\( T^{7} + \)\(50\!\cdots\!14\)\( T^{8} + \)\(81\!\cdots\!40\)\( T^{9} + \)\(21\!\cdots\!30\)\( T^{10} + \)\(38\!\cdots\!35\)\( T^{11} + \)\(78\!\cdots\!73\)\( T^{12} + \)\(14\!\cdots\!64\)\( T^{13} + \)\(25\!\cdots\!76\)\( T^{14} + \)\(48\!\cdots\!06\)\( T^{15} + \)\(72\!\cdots\!88\)\( T^{16} + \)\(48\!\cdots\!06\)\( p^{7} T^{17} + \)\(25\!\cdots\!76\)\( p^{14} T^{18} + \)\(14\!\cdots\!64\)\( p^{21} T^{19} + \)\(78\!\cdots\!73\)\( p^{28} T^{20} + \)\(38\!\cdots\!35\)\( p^{35} T^{21} + \)\(21\!\cdots\!30\)\( p^{42} T^{22} + \)\(81\!\cdots\!40\)\( p^{49} T^{23} + \)\(50\!\cdots\!14\)\( p^{56} T^{24} + \)\(11\!\cdots\!03\)\( p^{63} T^{25} + \)\(10\!\cdots\!22\)\( p^{70} T^{26} + \)\(59\!\cdots\!05\)\( p^{77} T^{27} + \)\(16\!\cdots\!06\)\( p^{84} T^{28} - \)\(12\!\cdots\!64\)\( p^{91} T^{29} + 177736299568080 p^{98} T^{30} - 1951425 p^{105} T^{31} + p^{112} T^{32} \)
89 \( 1 + 10499443 T + 426097160615769 T^{2} + \)\(42\!\cdots\!15\)\( T^{3} + \)\(91\!\cdots\!59\)\( T^{4} + \)\(86\!\cdots\!14\)\( T^{5} + \)\(13\!\cdots\!38\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!19\)\( T^{8} + \)\(11\!\cdots\!43\)\( T^{9} + \)\(11\!\cdots\!47\)\( T^{10} + \)\(86\!\cdots\!21\)\( T^{11} + \)\(78\!\cdots\!47\)\( T^{12} + \)\(54\!\cdots\!14\)\( T^{13} + \)\(44\!\cdots\!18\)\( T^{14} + \)\(28\!\cdots\!80\)\( T^{15} + \)\(21\!\cdots\!04\)\( T^{16} + \)\(28\!\cdots\!80\)\( p^{7} T^{17} + \)\(44\!\cdots\!18\)\( p^{14} T^{18} + \)\(54\!\cdots\!14\)\( p^{21} T^{19} + \)\(78\!\cdots\!47\)\( p^{28} T^{20} + \)\(86\!\cdots\!21\)\( p^{35} T^{21} + \)\(11\!\cdots\!47\)\( p^{42} T^{22} + \)\(11\!\cdots\!43\)\( p^{49} T^{23} + \)\(13\!\cdots\!19\)\( p^{56} T^{24} + \)\(11\!\cdots\!50\)\( p^{63} T^{25} + \)\(13\!\cdots\!38\)\( p^{70} T^{26} + \)\(86\!\cdots\!14\)\( p^{77} T^{27} + \)\(91\!\cdots\!59\)\( p^{84} T^{28} + \)\(42\!\cdots\!15\)\( p^{91} T^{29} + 426097160615769 p^{98} T^{30} + 10499443 p^{105} T^{31} + p^{112} T^{32} \)
97 \( 1 + 25158976 T + 9489803532067 p T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(36\!\cdots\!44\)\( T^{4} + \)\(55\!\cdots\!08\)\( T^{5} + \)\(91\!\cdots\!59\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} + \)\(16\!\cdots\!05\)\( T^{8} + \)\(18\!\cdots\!56\)\( T^{9} + \)\(23\!\cdots\!12\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} + \)\(27\!\cdots\!22\)\( T^{12} + \)\(26\!\cdots\!88\)\( T^{13} + \)\(27\!\cdots\!94\)\( T^{14} + \)\(25\!\cdots\!28\)\( T^{15} + \)\(24\!\cdots\!12\)\( T^{16} + \)\(25\!\cdots\!28\)\( p^{7} T^{17} + \)\(27\!\cdots\!94\)\( p^{14} T^{18} + \)\(26\!\cdots\!88\)\( p^{21} T^{19} + \)\(27\!\cdots\!22\)\( p^{28} T^{20} + \)\(24\!\cdots\!60\)\( p^{35} T^{21} + \)\(23\!\cdots\!12\)\( p^{42} T^{22} + \)\(18\!\cdots\!56\)\( p^{49} T^{23} + \)\(16\!\cdots\!05\)\( p^{56} T^{24} + \)\(11\!\cdots\!24\)\( p^{63} T^{25} + \)\(91\!\cdots\!59\)\( p^{70} T^{26} + \)\(55\!\cdots\!08\)\( p^{77} T^{27} + \)\(36\!\cdots\!44\)\( p^{84} T^{28} + \)\(17\!\cdots\!08\)\( p^{91} T^{29} + 9489803532067 p^{99} T^{30} + 25158976 p^{105} T^{31} + p^{112} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.46198419467559134548889482139, −2.29816933464404822591477315531, −2.24290977331084358854346045013, −2.22694172232322113070600665715, −2.18069913315920447863644747827, −2.04767650883440137915100891365, −1.98169019823110748198408245775, −1.88220034610236744903664526886, −1.85192107895209792244282561895, −1.76215566602864886638738493270, −1.72983141344295799680681636428, −1.55092586890205285450691649072, −1.49184665751187722813609008011, −1.36755121563122431001990874951, −1.34241343611899950859386596028, −1.33705043368291974427725887412, −1.26922795471849609815744761808, −1.23274980586705725350414018262, −1.20445961754760039631295392465, −1.07454455910264601714886975843, −0.933251052791349761453853512215, −0.913253144127034297947856632617, −0.818915492883988648642269893901, −0.78217434556804683614811557265, −0.53674707995036547970847649452, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.53674707995036547970847649452, 0.78217434556804683614811557265, 0.818915492883988648642269893901, 0.913253144127034297947856632617, 0.933251052791349761453853512215, 1.07454455910264601714886975843, 1.20445961754760039631295392465, 1.23274980586705725350414018262, 1.26922795471849609815744761808, 1.33705043368291974427725887412, 1.34241343611899950859386596028, 1.36755121563122431001990874951, 1.49184665751187722813609008011, 1.55092586890205285450691649072, 1.72983141344295799680681636428, 1.76215566602864886638738493270, 1.85192107895209792244282561895, 1.88220034610236744903664526886, 1.98169019823110748198408245775, 2.04767650883440137915100891365, 2.18069913315920447863644747827, 2.22694172232322113070600665715, 2.24290977331084358854346045013, 2.29816933464404822591477315531, 2.46198419467559134548889482139

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

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