Properties

Degree 30
Conductor $ 43^{15} $
Sign $-1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 15

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s − 317·3-s − 1.70e3·4-s − 4.71e3·5-s + 1.01e4·6-s − 9.68e3·7-s + 6.42e4·8-s − 6.26e4·9-s + 1.50e5·10-s − 1.04e5·11-s + 5.41e5·12-s − 1.16e5·13-s + 3.09e5·14-s + 1.49e6·15-s + 1.24e6·16-s − 8.84e5·17-s + 2.00e6·18-s − 6.89e5·19-s + 8.05e6·20-s + 3.06e6·21-s + 3.34e6·22-s − 2.50e6·23-s − 2.03e7·24-s − 2.86e6·25-s + 3.71e6·26-s + 2.83e7·27-s + 1.65e7·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.25·3-s − 3.33·4-s − 3.37·5-s + 3.19·6-s − 1.52·7-s + 5.54·8-s − 3.18·9-s + 4.77·10-s − 2.15·11-s + 7.53·12-s − 1.12·13-s + 2.15·14-s + 7.62·15-s + 4.75·16-s − 2.56·17-s + 4.49·18-s − 1.21·19-s + 11.2·20-s + 3.44·21-s + 3.04·22-s − 1.86·23-s − 12.5·24-s − 1.46·25-s + 1.59·26-s + 10.2·27-s + 5.08·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{15}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(30\)
\( N \)  =  \(43^{15}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(15\)
Selberg data  =  \((30,\ 43^{15} ,\ ( \ : [9/2]^{15} ),\ -1 )\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   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 30. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 29.
$p$$F_p(T)$
bad43 \( ( 1 + p^{4} T )^{15} \)
good2 \( 1 + p^{5} T + 2731 T^{2} + 38903 p T^{3} + 1925179 p T^{4} + 12729931 p^{3} T^{5} + 466748515 p^{3} T^{6} + 2790127665 p^{5} T^{7} + 42541932853 p^{6} T^{8} + 226560437741 p^{8} T^{9} + 3119560601683 p^{9} T^{10} + 14760976902969 p^{11} T^{11} + 48909112098911 p^{14} T^{12} + 213220815288751 p^{16} T^{13} + 2927267306494753 p^{17} T^{14} + 12819018469580755 p^{19} T^{15} + 2927267306494753 p^{26} T^{16} + 213220815288751 p^{34} T^{17} + 48909112098911 p^{41} T^{18} + 14760976902969 p^{47} T^{19} + 3119560601683 p^{54} T^{20} + 226560437741 p^{62} T^{21} + 42541932853 p^{69} T^{22} + 2790127665 p^{77} T^{23} + 466748515 p^{84} T^{24} + 12729931 p^{93} T^{25} + 1925179 p^{100} T^{26} + 38903 p^{109} T^{27} + 2731 p^{117} T^{28} + p^{131} T^{29} + p^{135} T^{30} \)
3 \( 1 + 317 T + 163109 T^{2} + 14396569 p T^{3} + 13657333057 T^{4} + 3049240516432 T^{5} + 749507085253976 T^{6} + 1815939834537548 p^{4} T^{7} + 1130216186040245534 p^{3} T^{8} + \)\(19\!\cdots\!38\)\( p^{3} T^{9} + \)\(40\!\cdots\!23\)\( p^{5} T^{10} + \)\(21\!\cdots\!09\)\( p^{6} T^{11} + \)\(36\!\cdots\!85\)\( p^{6} T^{12} + \)\(60\!\cdots\!61\)\( p^{8} T^{13} + \)\(30\!\cdots\!53\)\( p^{9} T^{14} + \)\(42\!\cdots\!40\)\( p^{9} T^{15} + \)\(30\!\cdots\!53\)\( p^{18} T^{16} + \)\(60\!\cdots\!61\)\( p^{26} T^{17} + \)\(36\!\cdots\!85\)\( p^{33} T^{18} + \)\(21\!\cdots\!09\)\( p^{42} T^{19} + \)\(40\!\cdots\!23\)\( p^{50} T^{20} + \)\(19\!\cdots\!38\)\( p^{57} T^{21} + 1130216186040245534 p^{66} T^{22} + 1815939834537548 p^{76} T^{23} + 749507085253976 p^{81} T^{24} + 3049240516432 p^{90} T^{25} + 13657333057 p^{99} T^{26} + 14396569 p^{109} T^{27} + 163109 p^{117} T^{28} + 317 p^{126} T^{29} + p^{135} T^{30} \)
5 \( 1 + 4717 T + 25115807 T^{2} + 83707229397 T^{3} + 275195357593257 T^{4} + 146051295003353118 p T^{5} + \)\(18\!\cdots\!78\)\( T^{6} + \)\(41\!\cdots\!18\)\( T^{7} + \)\(89\!\cdots\!28\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(32\!\cdots\!59\)\( T^{10} + \)\(11\!\cdots\!91\)\( p T^{11} + \)\(38\!\cdots\!17\)\( p^{2} T^{12} + \)\(12\!\cdots\!89\)\( p^{3} T^{13} + \)\(36\!\cdots\!97\)\( p^{4} T^{14} + \)\(10\!\cdots\!08\)\( p^{5} T^{15} + \)\(36\!\cdots\!97\)\( p^{13} T^{16} + \)\(12\!\cdots\!89\)\( p^{21} T^{17} + \)\(38\!\cdots\!17\)\( p^{29} T^{18} + \)\(11\!\cdots\!91\)\( p^{37} T^{19} + \)\(32\!\cdots\!59\)\( p^{45} T^{20} + \)\(17\!\cdots\!68\)\( p^{54} T^{21} + \)\(89\!\cdots\!28\)\( p^{63} T^{22} + \)\(41\!\cdots\!18\)\( p^{72} T^{23} + \)\(18\!\cdots\!78\)\( p^{81} T^{24} + 146051295003353118 p^{91} T^{25} + 275195357593257 p^{99} T^{26} + 83707229397 p^{108} T^{27} + 25115807 p^{117} T^{28} + 4717 p^{126} T^{29} + p^{135} T^{30} \)
7 \( 1 + 9680 T + 303171705 T^{2} + 2420677168232 T^{3} + 5969978681963587 p T^{4} + \)\(27\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!17\)\( T^{6} + \)\(25\!\cdots\!12\)\( p T^{7} + \)\(37\!\cdots\!89\)\( p^{2} T^{8} + \)\(18\!\cdots\!52\)\( p^{3} T^{9} + \)\(24\!\cdots\!33\)\( p^{4} T^{10} - \)\(69\!\cdots\!88\)\( p^{6} T^{11} + \)\(45\!\cdots\!97\)\( p^{6} T^{12} - \)\(17\!\cdots\!52\)\( p^{7} T^{13} - \)\(76\!\cdots\!71\)\( p^{8} T^{14} - \)\(29\!\cdots\!84\)\( p^{10} T^{15} - \)\(76\!\cdots\!71\)\( p^{17} T^{16} - \)\(17\!\cdots\!52\)\( p^{25} T^{17} + \)\(45\!\cdots\!97\)\( p^{33} T^{18} - \)\(69\!\cdots\!88\)\( p^{42} T^{19} + \)\(24\!\cdots\!33\)\( p^{49} T^{20} + \)\(18\!\cdots\!52\)\( p^{57} T^{21} + \)\(37\!\cdots\!89\)\( p^{65} T^{22} + \)\(25\!\cdots\!12\)\( p^{73} T^{23} + \)\(34\!\cdots\!17\)\( p^{81} T^{24} + \)\(27\!\cdots\!60\)\( p^{90} T^{25} + 5969978681963587 p^{100} T^{26} + 2420677168232 p^{108} T^{27} + 303171705 p^{117} T^{28} + 9680 p^{126} T^{29} + p^{135} T^{30} \)
11 \( 1 + 104484 T + 21671047097 T^{2} + 1826154580412208 T^{3} + \)\(22\!\cdots\!79\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!71\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(81\!\cdots\!64\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(29\!\cdots\!64\)\( p T^{10} + \)\(17\!\cdots\!28\)\( T^{11} + \)\(10\!\cdots\!38\)\( T^{12} + \)\(52\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!22\)\( p T^{14} + \)\(13\!\cdots\!92\)\( T^{15} + \)\(27\!\cdots\!22\)\( p^{10} T^{16} + \)\(52\!\cdots\!68\)\( p^{18} T^{17} + \)\(10\!\cdots\!38\)\( p^{27} T^{18} + \)\(17\!\cdots\!28\)\( p^{36} T^{19} + \)\(29\!\cdots\!64\)\( p^{46} T^{20} + \)\(46\!\cdots\!00\)\( p^{54} T^{21} + \)\(81\!\cdots\!64\)\( p^{63} T^{22} + \)\(10\!\cdots\!88\)\( p^{72} T^{23} + \)\(15\!\cdots\!71\)\( p^{81} T^{24} + \)\(16\!\cdots\!04\)\( p^{90} T^{25} + \)\(22\!\cdots\!79\)\( p^{99} T^{26} + 1826154580412208 p^{108} T^{27} + 21671047097 p^{117} T^{28} + 104484 p^{126} T^{29} + p^{135} T^{30} \)
13 \( 1 + 116174 T + 93110646219 T^{2} + 9939551633003756 T^{3} + \)\(33\!\cdots\!91\)\( p T^{4} + \)\(42\!\cdots\!54\)\( T^{5} + \)\(13\!\cdots\!41\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!88\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!96\)\( T^{10} + \)\(42\!\cdots\!04\)\( T^{11} + \)\(85\!\cdots\!78\)\( T^{12} + \)\(58\!\cdots\!52\)\( T^{13} + \)\(10\!\cdots\!26\)\( T^{14} + \)\(67\!\cdots\!12\)\( T^{15} + \)\(10\!\cdots\!26\)\( p^{9} T^{16} + \)\(58\!\cdots\!52\)\( p^{18} T^{17} + \)\(85\!\cdots\!78\)\( p^{27} T^{18} + \)\(42\!\cdots\!04\)\( p^{36} T^{19} + \)\(57\!\cdots\!96\)\( p^{45} T^{20} + \)\(25\!\cdots\!00\)\( p^{54} T^{21} + \)\(31\!\cdots\!88\)\( p^{63} T^{22} + \)\(12\!\cdots\!40\)\( p^{72} T^{23} + \)\(13\!\cdots\!41\)\( p^{81} T^{24} + \)\(42\!\cdots\!54\)\( p^{90} T^{25} + \)\(33\!\cdots\!91\)\( p^{100} T^{26} + 9939551633003756 p^{108} T^{27} + 93110646219 p^{117} T^{28} + 116174 p^{126} T^{29} + p^{135} T^{30} \)
17 \( 1 + 884265 T + 1384046763853 T^{2} + 1008991515705553461 T^{3} + \)\(92\!\cdots\!03\)\( T^{4} + \)\(56\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!52\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(54\!\cdots\!13\)\( T^{9} + \)\(26\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!12\)\( T^{11} + \)\(47\!\cdots\!38\)\( T^{12} + \)\(17\!\cdots\!77\)\( T^{13} + \)\(69\!\cdots\!95\)\( T^{14} + \)\(23\!\cdots\!62\)\( T^{15} + \)\(69\!\cdots\!95\)\( p^{9} T^{16} + \)\(17\!\cdots\!77\)\( p^{18} T^{17} + \)\(47\!\cdots\!38\)\( p^{27} T^{18} + \)\(11\!\cdots\!12\)\( p^{36} T^{19} + \)\(26\!\cdots\!40\)\( p^{45} T^{20} + \)\(54\!\cdots\!13\)\( p^{54} T^{21} + \)\(11\!\cdots\!15\)\( p^{63} T^{22} + \)\(20\!\cdots\!80\)\( p^{72} T^{23} + \)\(38\!\cdots\!52\)\( p^{81} T^{24} + \)\(56\!\cdots\!24\)\( p^{90} T^{25} + \)\(92\!\cdots\!03\)\( p^{99} T^{26} + 1008991515705553461 p^{108} T^{27} + 1384046763853 p^{117} T^{28} + 884265 p^{126} T^{29} + p^{135} T^{30} \)
19 \( 1 + 689535 T + 2651913177639 T^{2} + 1907898195520184901 T^{3} + \)\(36\!\cdots\!69\)\( T^{4} + \)\(26\!\cdots\!68\)\( T^{5} + \)\(34\!\cdots\!48\)\( T^{6} + \)\(13\!\cdots\!96\)\( p T^{7} + \)\(24\!\cdots\!78\)\( T^{8} + \)\(17\!\cdots\!10\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} + \)\(48\!\cdots\!09\)\( p T^{11} + \)\(66\!\cdots\!35\)\( T^{12} + \)\(40\!\cdots\!15\)\( T^{13} + \)\(25\!\cdots\!19\)\( T^{14} + \)\(14\!\cdots\!88\)\( T^{15} + \)\(25\!\cdots\!19\)\( p^{9} T^{16} + \)\(40\!\cdots\!15\)\( p^{18} T^{17} + \)\(66\!\cdots\!35\)\( p^{27} T^{18} + \)\(48\!\cdots\!09\)\( p^{37} T^{19} + \)\(14\!\cdots\!57\)\( p^{45} T^{20} + \)\(17\!\cdots\!10\)\( p^{54} T^{21} + \)\(24\!\cdots\!78\)\( p^{63} T^{22} + \)\(13\!\cdots\!96\)\( p^{73} T^{23} + \)\(34\!\cdots\!48\)\( p^{81} T^{24} + \)\(26\!\cdots\!68\)\( p^{90} T^{25} + \)\(36\!\cdots\!69\)\( p^{99} T^{26} + 1907898195520184901 p^{108} T^{27} + 2651913177639 p^{117} T^{28} + 689535 p^{126} T^{29} + p^{135} T^{30} \)
23 \( 1 + 2504077 T + 862494392727 p T^{2} + 44904598844137902367 T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!74\)\( T^{5} + \)\(12\!\cdots\!34\)\( T^{6} + \)\(22\!\cdots\!78\)\( T^{7} + \)\(54\!\cdots\!85\)\( T^{8} + \)\(92\!\cdots\!11\)\( T^{9} + \)\(19\!\cdots\!50\)\( T^{10} + \)\(29\!\cdots\!96\)\( T^{11} + \)\(52\!\cdots\!24\)\( T^{12} + \)\(73\!\cdots\!07\)\( T^{13} + \)\(11\!\cdots\!35\)\( T^{14} + \)\(14\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!35\)\( p^{9} T^{16} + \)\(73\!\cdots\!07\)\( p^{18} T^{17} + \)\(52\!\cdots\!24\)\( p^{27} T^{18} + \)\(29\!\cdots\!96\)\( p^{36} T^{19} + \)\(19\!\cdots\!50\)\( p^{45} T^{20} + \)\(92\!\cdots\!11\)\( p^{54} T^{21} + \)\(54\!\cdots\!85\)\( p^{63} T^{22} + \)\(22\!\cdots\!78\)\( p^{72} T^{23} + \)\(12\!\cdots\!34\)\( p^{81} T^{24} + \)\(39\!\cdots\!74\)\( p^{90} T^{25} + \)\(19\!\cdots\!31\)\( p^{99} T^{26} + 44904598844137902367 p^{108} T^{27} + 862494392727 p^{118} T^{28} + 2504077 p^{126} T^{29} + p^{135} T^{30} \)
29 \( 1 + 18406221 T + 231322256139583 T^{2} + \)\(21\!\cdots\!69\)\( T^{3} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!86\)\( T^{5} + \)\(64\!\cdots\!34\)\( T^{6} + \)\(34\!\cdots\!58\)\( T^{7} + \)\(16\!\cdots\!04\)\( T^{8} + \)\(73\!\cdots\!72\)\( T^{9} + \)\(30\!\cdots\!31\)\( T^{10} + \)\(11\!\cdots\!11\)\( T^{11} + \)\(44\!\cdots\!21\)\( T^{12} + \)\(16\!\cdots\!53\)\( T^{13} + \)\(60\!\cdots\!37\)\( T^{14} + \)\(22\!\cdots\!12\)\( T^{15} + \)\(60\!\cdots\!37\)\( p^{9} T^{16} + \)\(16\!\cdots\!53\)\( p^{18} T^{17} + \)\(44\!\cdots\!21\)\( p^{27} T^{18} + \)\(11\!\cdots\!11\)\( p^{36} T^{19} + \)\(30\!\cdots\!31\)\( p^{45} T^{20} + \)\(73\!\cdots\!72\)\( p^{54} T^{21} + \)\(16\!\cdots\!04\)\( p^{63} T^{22} + \)\(34\!\cdots\!58\)\( p^{72} T^{23} + \)\(64\!\cdots\!34\)\( p^{81} T^{24} + \)\(10\!\cdots\!86\)\( p^{90} T^{25} + \)\(16\!\cdots\!45\)\( p^{99} T^{26} + \)\(21\!\cdots\!69\)\( p^{108} T^{27} + 231322256139583 p^{117} T^{28} + 18406221 p^{126} T^{29} + p^{135} T^{30} \)
31 \( 1 + 12033699 T + 157221027222111 T^{2} + \)\(11\!\cdots\!53\)\( T^{3} + \)\(11\!\cdots\!51\)\( T^{4} + \)\(78\!\cdots\!70\)\( T^{5} + \)\(65\!\cdots\!70\)\( T^{6} + \)\(42\!\cdots\!02\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} + \)\(18\!\cdots\!53\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} + \)\(68\!\cdots\!12\)\( T^{11} + \)\(39\!\cdots\!74\)\( T^{12} + \)\(21\!\cdots\!69\)\( T^{13} + \)\(11\!\cdots\!43\)\( T^{14} + \)\(62\!\cdots\!76\)\( T^{15} + \)\(11\!\cdots\!43\)\( p^{9} T^{16} + \)\(21\!\cdots\!69\)\( p^{18} T^{17} + \)\(39\!\cdots\!74\)\( p^{27} T^{18} + \)\(68\!\cdots\!12\)\( p^{36} T^{19} + \)\(11\!\cdots\!80\)\( p^{45} T^{20} + \)\(18\!\cdots\!53\)\( p^{54} T^{21} + \)\(29\!\cdots\!41\)\( p^{63} T^{22} + \)\(42\!\cdots\!02\)\( p^{72} T^{23} + \)\(65\!\cdots\!70\)\( p^{81} T^{24} + \)\(78\!\cdots\!70\)\( p^{90} T^{25} + \)\(11\!\cdots\!51\)\( p^{99} T^{26} + \)\(11\!\cdots\!53\)\( p^{108} T^{27} + 157221027222111 p^{117} T^{28} + 12033699 p^{126} T^{29} + p^{135} T^{30} \)
37 \( 1 + 8722847 T + 568554278402447 T^{2} + \)\(60\!\cdots\!91\)\( T^{3} + \)\(19\!\cdots\!13\)\( T^{4} + \)\(18\!\cdots\!10\)\( T^{5} + \)\(47\!\cdots\!42\)\( T^{6} + \)\(39\!\cdots\!30\)\( T^{7} + \)\(80\!\cdots\!88\)\( T^{8} + \)\(52\!\cdots\!84\)\( T^{9} + \)\(99\!\cdots\!43\)\( T^{10} + \)\(39\!\cdots\!97\)\( T^{11} + \)\(88\!\cdots\!13\)\( T^{12} - \)\(31\!\cdots\!73\)\( T^{13} + \)\(63\!\cdots\!65\)\( T^{14} - \)\(36\!\cdots\!60\)\( T^{15} + \)\(63\!\cdots\!65\)\( p^{9} T^{16} - \)\(31\!\cdots\!73\)\( p^{18} T^{17} + \)\(88\!\cdots\!13\)\( p^{27} T^{18} + \)\(39\!\cdots\!97\)\( p^{36} T^{19} + \)\(99\!\cdots\!43\)\( p^{45} T^{20} + \)\(52\!\cdots\!84\)\( p^{54} T^{21} + \)\(80\!\cdots\!88\)\( p^{63} T^{22} + \)\(39\!\cdots\!30\)\( p^{72} T^{23} + \)\(47\!\cdots\!42\)\( p^{81} T^{24} + \)\(18\!\cdots\!10\)\( p^{90} T^{25} + \)\(19\!\cdots\!13\)\( p^{99} T^{26} + \)\(60\!\cdots\!91\)\( p^{108} T^{27} + 568554278402447 p^{117} T^{28} + 8722847 p^{126} T^{29} + p^{135} T^{30} \)
41 \( 1 + 18689389 T + 2776615386417537 T^{2} + \)\(46\!\cdots\!33\)\( T^{3} + \)\(91\!\cdots\!39\)\( p T^{4} + \)\(58\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!84\)\( T^{6} + \)\(48\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!31\)\( T^{8} + \)\(30\!\cdots\!37\)\( T^{9} + \)\(11\!\cdots\!84\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(48\!\cdots\!18\)\( T^{12} + \)\(63\!\cdots\!65\)\( T^{13} + \)\(18\!\cdots\!43\)\( T^{14} + \)\(22\!\cdots\!26\)\( T^{15} + \)\(18\!\cdots\!43\)\( p^{9} T^{16} + \)\(63\!\cdots\!65\)\( p^{18} T^{17} + \)\(48\!\cdots\!18\)\( p^{27} T^{18} + \)\(15\!\cdots\!00\)\( p^{36} T^{19} + \)\(11\!\cdots\!84\)\( p^{45} T^{20} + \)\(30\!\cdots\!37\)\( p^{54} T^{21} + \)\(21\!\cdots\!31\)\( p^{63} T^{22} + \)\(48\!\cdots\!36\)\( p^{72} T^{23} + \)\(33\!\cdots\!84\)\( p^{81} T^{24} + \)\(58\!\cdots\!96\)\( p^{90} T^{25} + \)\(91\!\cdots\!39\)\( p^{100} T^{26} + \)\(46\!\cdots\!33\)\( p^{108} T^{27} + 2776615386417537 p^{117} T^{28} + 18689389 p^{126} T^{29} + p^{135} T^{30} \)
47 \( 1 + 104960741 T + 13310159754260471 T^{2} + \)\(95\!\cdots\!91\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} + \)\(92\!\cdots\!28\)\( p T^{5} + \)\(26\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!76\)\( T^{7} + \)\(66\!\cdots\!90\)\( T^{8} + \)\(28\!\cdots\!38\)\( T^{9} + \)\(12\!\cdots\!89\)\( T^{10} + \)\(49\!\cdots\!97\)\( T^{11} + \)\(20\!\cdots\!91\)\( T^{12} + \)\(70\!\cdots\!13\)\( T^{13} + \)\(26\!\cdots\!11\)\( T^{14} + \)\(85\!\cdots\!24\)\( T^{15} + \)\(26\!\cdots\!11\)\( p^{9} T^{16} + \)\(70\!\cdots\!13\)\( p^{18} T^{17} + \)\(20\!\cdots\!91\)\( p^{27} T^{18} + \)\(49\!\cdots\!97\)\( p^{36} T^{19} + \)\(12\!\cdots\!89\)\( p^{45} T^{20} + \)\(28\!\cdots\!38\)\( p^{54} T^{21} + \)\(66\!\cdots\!90\)\( p^{63} T^{22} + \)\(13\!\cdots\!76\)\( p^{72} T^{23} + \)\(26\!\cdots\!08\)\( p^{81} T^{24} + \)\(92\!\cdots\!28\)\( p^{91} T^{25} + \)\(75\!\cdots\!21\)\( p^{99} T^{26} + \)\(95\!\cdots\!91\)\( p^{108} T^{27} + 13310159754260471 p^{117} T^{28} + 104960741 p^{126} T^{29} + p^{135} T^{30} \)
53 \( 1 + 215907800 T + 45601589020862355 T^{2} + \)\(63\!\cdots\!44\)\( T^{3} + \)\(82\!\cdots\!07\)\( T^{4} + \)\(85\!\cdots\!20\)\( T^{5} + \)\(82\!\cdots\!01\)\( T^{6} + \)\(67\!\cdots\!72\)\( T^{7} + \)\(51\!\cdots\!60\)\( T^{8} + \)\(34\!\cdots\!76\)\( T^{9} + \)\(21\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{11} + \)\(58\!\cdots\!90\)\( T^{12} + \)\(26\!\cdots\!88\)\( T^{13} + \)\(12\!\cdots\!62\)\( T^{14} + \)\(61\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!62\)\( p^{9} T^{16} + \)\(26\!\cdots\!88\)\( p^{18} T^{17} + \)\(58\!\cdots\!90\)\( p^{27} T^{18} + \)\(11\!\cdots\!24\)\( p^{36} T^{19} + \)\(21\!\cdots\!24\)\( p^{45} T^{20} + \)\(34\!\cdots\!76\)\( p^{54} T^{21} + \)\(51\!\cdots\!60\)\( p^{63} T^{22} + \)\(67\!\cdots\!72\)\( p^{72} T^{23} + \)\(82\!\cdots\!01\)\( p^{81} T^{24} + \)\(85\!\cdots\!20\)\( p^{90} T^{25} + \)\(82\!\cdots\!07\)\( p^{99} T^{26} + \)\(63\!\cdots\!44\)\( p^{108} T^{27} + 45601589020862355 p^{117} T^{28} + 215907800 p^{126} T^{29} + p^{135} T^{30} \)
59 \( 1 - 185924544 T + 72641643327616985 T^{2} - \)\(11\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!25\)\( T^{4} - \)\(37\!\cdots\!76\)\( T^{5} + \)\(67\!\cdots\!65\)\( T^{6} - \)\(85\!\cdots\!64\)\( T^{7} + \)\(12\!\cdots\!49\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!45\)\( T^{10} - \)\(20\!\cdots\!16\)\( T^{11} + \)\(24\!\cdots\!49\)\( T^{12} - \)\(23\!\cdots\!92\)\( T^{13} + \)\(25\!\cdots\!69\)\( T^{14} - \)\(22\!\cdots\!84\)\( T^{15} + \)\(25\!\cdots\!69\)\( p^{9} T^{16} - \)\(23\!\cdots\!92\)\( p^{18} T^{17} + \)\(24\!\cdots\!49\)\( p^{27} T^{18} - \)\(20\!\cdots\!16\)\( p^{36} T^{19} + \)\(19\!\cdots\!45\)\( p^{45} T^{20} - \)\(14\!\cdots\!84\)\( p^{54} T^{21} + \)\(12\!\cdots\!49\)\( p^{63} T^{22} - \)\(85\!\cdots\!64\)\( p^{72} T^{23} + \)\(67\!\cdots\!65\)\( p^{81} T^{24} - \)\(37\!\cdots\!76\)\( p^{90} T^{25} + \)\(26\!\cdots\!25\)\( p^{99} T^{26} - \)\(11\!\cdots\!56\)\( p^{108} T^{27} + 72641643327616985 p^{117} T^{28} - 185924544 p^{126} T^{29} + p^{135} T^{30} \)
61 \( 1 - 247538102 T + 123939924002267915 T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(72\!\cdots\!53\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} - \)\(42\!\cdots\!04\)\( T^{7} + \)\(73\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(15\!\cdots\!63\)\( T^{10} - \)\(19\!\cdots\!64\)\( T^{11} + \)\(26\!\cdots\!33\)\( T^{12} - \)\(30\!\cdots\!38\)\( T^{13} + \)\(37\!\cdots\!95\)\( T^{14} - \)\(39\!\cdots\!08\)\( T^{15} + \)\(37\!\cdots\!95\)\( p^{9} T^{16} - \)\(30\!\cdots\!38\)\( p^{18} T^{17} + \)\(26\!\cdots\!33\)\( p^{27} T^{18} - \)\(19\!\cdots\!64\)\( p^{36} T^{19} + \)\(15\!\cdots\!63\)\( p^{45} T^{20} - \)\(10\!\cdots\!70\)\( p^{54} T^{21} + \)\(73\!\cdots\!05\)\( p^{63} T^{22} - \)\(42\!\cdots\!04\)\( p^{72} T^{23} + \)\(27\!\cdots\!23\)\( p^{81} T^{24} - \)\(12\!\cdots\!30\)\( p^{90} T^{25} + \)\(72\!\cdots\!53\)\( p^{99} T^{26} - \)\(25\!\cdots\!76\)\( p^{108} T^{27} + 123939924002267915 p^{117} T^{28} - 247538102 p^{126} T^{29} + p^{135} T^{30} \)
67 \( 1 - 467904656 T + 289992155760016797 T^{2} - \)\(96\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!55\)\( T^{4} - \)\(99\!\cdots\!68\)\( T^{5} + \)\(29\!\cdots\!91\)\( T^{6} - \)\(69\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} - \)\(36\!\cdots\!76\)\( T^{9} + \)\(81\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(31\!\cdots\!82\)\( T^{12} - \)\(54\!\cdots\!16\)\( T^{13} + \)\(10\!\cdots\!46\)\( T^{14} - \)\(16\!\cdots\!44\)\( T^{15} + \)\(10\!\cdots\!46\)\( p^{9} T^{16} - \)\(54\!\cdots\!16\)\( p^{18} T^{17} + \)\(31\!\cdots\!82\)\( p^{27} T^{18} - \)\(15\!\cdots\!04\)\( p^{36} T^{19} + \)\(81\!\cdots\!48\)\( p^{45} T^{20} - \)\(36\!\cdots\!76\)\( p^{54} T^{21} + \)\(17\!\cdots\!04\)\( p^{63} T^{22} - \)\(69\!\cdots\!20\)\( p^{72} T^{23} + \)\(29\!\cdots\!91\)\( p^{81} T^{24} - \)\(99\!\cdots\!68\)\( p^{90} T^{25} + \)\(36\!\cdots\!55\)\( p^{99} T^{26} - \)\(96\!\cdots\!32\)\( p^{108} T^{27} + 289992155760016797 p^{117} T^{28} - 467904656 p^{126} T^{29} + p^{135} T^{30} \)
71 \( 1 + 8252944 T + 295175699032282861 T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!81\)\( T^{4} + \)\(31\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!57\)\( T^{6} + \)\(47\!\cdots\!60\)\( T^{7} + \)\(40\!\cdots\!77\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} + \)\(28\!\cdots\!97\)\( T^{10} + \)\(35\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!41\)\( T^{12} + \)\(20\!\cdots\!76\)\( T^{13} + \)\(94\!\cdots\!13\)\( T^{14} + \)\(10\!\cdots\!84\)\( T^{15} + \)\(94\!\cdots\!13\)\( p^{9} T^{16} + \)\(20\!\cdots\!76\)\( p^{18} T^{17} + \)\(17\!\cdots\!41\)\( p^{27} T^{18} + \)\(35\!\cdots\!60\)\( p^{36} T^{19} + \)\(28\!\cdots\!97\)\( p^{45} T^{20} + \)\(47\!\cdots\!80\)\( p^{54} T^{21} + \)\(40\!\cdots\!77\)\( p^{63} T^{22} + \)\(47\!\cdots\!60\)\( p^{72} T^{23} + \)\(48\!\cdots\!57\)\( p^{81} T^{24} + \)\(31\!\cdots\!80\)\( p^{90} T^{25} + \)\(45\!\cdots\!81\)\( p^{99} T^{26} + \)\(11\!\cdots\!80\)\( p^{108} T^{27} + 295175699032282861 p^{117} T^{28} + 8252944 p^{126} T^{29} + p^{135} T^{30} \)
73 \( 1 + 715627902 T + 725896042339854143 T^{2} + \)\(39\!\cdots\!04\)\( T^{3} + \)\(23\!\cdots\!09\)\( T^{4} + \)\(10\!\cdots\!42\)\( T^{5} + \)\(48\!\cdots\!43\)\( T^{6} + \)\(18\!\cdots\!64\)\( T^{7} + \)\(69\!\cdots\!09\)\( T^{8} + \)\(23\!\cdots\!46\)\( T^{9} + \)\(77\!\cdots\!47\)\( T^{10} + \)\(23\!\cdots\!12\)\( T^{11} + \)\(68\!\cdots\!33\)\( T^{12} + \)\(18\!\cdots\!86\)\( T^{13} + \)\(49\!\cdots\!39\)\( T^{14} + \)\(11\!\cdots\!52\)\( T^{15} + \)\(49\!\cdots\!39\)\( p^{9} T^{16} + \)\(18\!\cdots\!86\)\( p^{18} T^{17} + \)\(68\!\cdots\!33\)\( p^{27} T^{18} + \)\(23\!\cdots\!12\)\( p^{36} T^{19} + \)\(77\!\cdots\!47\)\( p^{45} T^{20} + \)\(23\!\cdots\!46\)\( p^{54} T^{21} + \)\(69\!\cdots\!09\)\( p^{63} T^{22} + \)\(18\!\cdots\!64\)\( p^{72} T^{23} + \)\(48\!\cdots\!43\)\( p^{81} T^{24} + \)\(10\!\cdots\!42\)\( p^{90} T^{25} + \)\(23\!\cdots\!09\)\( p^{99} T^{26} + \)\(39\!\cdots\!04\)\( p^{108} T^{27} + 725896042339854143 p^{117} T^{28} + 715627902 p^{126} T^{29} + p^{135} T^{30} \)
79 \( 1 - 560681783 T + 849133708768547829 T^{2} - \)\(32\!\cdots\!85\)\( T^{3} + \)\(33\!\cdots\!49\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(90\!\cdots\!32\)\( T^{6} - \)\(23\!\cdots\!28\)\( T^{7} + \)\(19\!\cdots\!98\)\( T^{8} - \)\(43\!\cdots\!74\)\( T^{9} + \)\(34\!\cdots\!01\)\( T^{10} - \)\(67\!\cdots\!51\)\( T^{11} + \)\(53\!\cdots\!45\)\( T^{12} - \)\(93\!\cdots\!63\)\( T^{13} + \)\(72\!\cdots\!51\)\( T^{14} - \)\(11\!\cdots\!88\)\( T^{15} + \)\(72\!\cdots\!51\)\( p^{9} T^{16} - \)\(93\!\cdots\!63\)\( p^{18} T^{17} + \)\(53\!\cdots\!45\)\( p^{27} T^{18} - \)\(67\!\cdots\!51\)\( p^{36} T^{19} + \)\(34\!\cdots\!01\)\( p^{45} T^{20} - \)\(43\!\cdots\!74\)\( p^{54} T^{21} + \)\(19\!\cdots\!98\)\( p^{63} T^{22} - \)\(23\!\cdots\!28\)\( p^{72} T^{23} + \)\(90\!\cdots\!32\)\( p^{81} T^{24} - \)\(10\!\cdots\!12\)\( p^{90} T^{25} + \)\(33\!\cdots\!49\)\( p^{99} T^{26} - \)\(32\!\cdots\!85\)\( p^{108} T^{27} + 849133708768547829 p^{117} T^{28} - 560681783 p^{126} T^{29} + p^{135} T^{30} \)
83 \( 1 + 1442854698 T + 2517006525957293097 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(27\!\cdots\!99\)\( T^{4} + \)\(22\!\cdots\!86\)\( T^{5} + \)\(19\!\cdots\!63\)\( T^{6} + \)\(13\!\cdots\!56\)\( T^{7} + \)\(92\!\cdots\!28\)\( T^{8} + \)\(56\!\cdots\!76\)\( T^{9} + \)\(33\!\cdots\!68\)\( T^{10} + \)\(18\!\cdots\!28\)\( T^{11} + \)\(97\!\cdots\!54\)\( T^{12} + \)\(47\!\cdots\!68\)\( T^{13} + \)\(22\!\cdots\!94\)\( T^{14} + \)\(98\!\cdots\!20\)\( T^{15} + \)\(22\!\cdots\!94\)\( p^{9} T^{16} + \)\(47\!\cdots\!68\)\( p^{18} T^{17} + \)\(97\!\cdots\!54\)\( p^{27} T^{18} + \)\(18\!\cdots\!28\)\( p^{36} T^{19} + \)\(33\!\cdots\!68\)\( p^{45} T^{20} + \)\(56\!\cdots\!76\)\( p^{54} T^{21} + \)\(92\!\cdots\!28\)\( p^{63} T^{22} + \)\(13\!\cdots\!56\)\( p^{72} T^{23} + \)\(19\!\cdots\!63\)\( p^{81} T^{24} + \)\(22\!\cdots\!86\)\( p^{90} T^{25} + \)\(27\!\cdots\!99\)\( p^{99} T^{26} + \)\(25\!\cdots\!08\)\( p^{108} T^{27} + 2517006525957293097 p^{117} T^{28} + 1442854698 p^{126} T^{29} + p^{135} T^{30} \)
89 \( 1 + 396710008 T + 2452763342866776295 T^{2} + \)\(86\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!85\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!31\)\( T^{6} + \)\(52\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!05\)\( T^{8} + \)\(17\!\cdots\!32\)\( T^{9} + \)\(51\!\cdots\!71\)\( T^{10} + \)\(14\!\cdots\!24\)\( T^{11} + \)\(17\!\cdots\!37\)\( T^{12} - \)\(18\!\cdots\!80\)\( T^{13} + \)\(54\!\cdots\!11\)\( T^{14} - \)\(10\!\cdots\!56\)\( T^{15} + \)\(54\!\cdots\!11\)\( p^{9} T^{16} - \)\(18\!\cdots\!80\)\( p^{18} T^{17} + \)\(17\!\cdots\!37\)\( p^{27} T^{18} + \)\(14\!\cdots\!24\)\( p^{36} T^{19} + \)\(51\!\cdots\!71\)\( p^{45} T^{20} + \)\(17\!\cdots\!32\)\( p^{54} T^{21} + \)\(12\!\cdots\!05\)\( p^{63} T^{22} + \)\(52\!\cdots\!68\)\( p^{72} T^{23} + \)\(22\!\cdots\!31\)\( p^{81} T^{24} + \)\(87\!\cdots\!92\)\( p^{90} T^{25} + \)\(29\!\cdots\!85\)\( p^{99} T^{26} + \)\(86\!\cdots\!68\)\( p^{108} T^{27} + 2452763342866776295 p^{117} T^{28} + 396710008 p^{126} T^{29} + p^{135} T^{30} \)
97 \( 1 + 3063837815 T + 9536233785462653481 T^{2} + \)\(19\!\cdots\!71\)\( T^{3} + \)\(38\!\cdots\!99\)\( T^{4} + \)\(61\!\cdots\!48\)\( T^{5} + \)\(97\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!04\)\( T^{7} + \)\(18\!\cdots\!19\)\( T^{8} + \)\(21\!\cdots\!95\)\( T^{9} + \)\(25\!\cdots\!96\)\( T^{10} + \)\(27\!\cdots\!24\)\( T^{11} + \)\(29\!\cdots\!54\)\( T^{12} + \)\(28\!\cdots\!39\)\( T^{13} + \)\(26\!\cdots\!31\)\( T^{14} + \)\(23\!\cdots\!02\)\( T^{15} + \)\(26\!\cdots\!31\)\( p^{9} T^{16} + \)\(28\!\cdots\!39\)\( p^{18} T^{17} + \)\(29\!\cdots\!54\)\( p^{27} T^{18} + \)\(27\!\cdots\!24\)\( p^{36} T^{19} + \)\(25\!\cdots\!96\)\( p^{45} T^{20} + \)\(21\!\cdots\!95\)\( p^{54} T^{21} + \)\(18\!\cdots\!19\)\( p^{63} T^{22} + \)\(13\!\cdots\!04\)\( p^{72} T^{23} + \)\(97\!\cdots\!08\)\( p^{81} T^{24} + \)\(61\!\cdots\!48\)\( p^{90} T^{25} + \)\(38\!\cdots\!99\)\( p^{99} T^{26} + \)\(19\!\cdots\!71\)\( p^{108} T^{27} + 9536233785462653481 p^{117} T^{28} + 3063837815 p^{126} T^{29} + p^{135} T^{30} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.00311384923013249688251675871, −3.87890453550281960566577088392, −3.77904499124625568549472568998, −3.73024449667682994690939444576, −3.71647038354970751297885216811, −3.61671572173132762865312158428, −3.54302431425833283025794108926, −3.52883635910927260172430903801, −3.10807836138298163644508610785, −2.97982993207549499072750346163, −2.89510699653770513205567736260, −2.82851255802725079238259674345, −2.60399065431517269471434369553, −2.58181485422625611206236865058, −2.35468996523677705203841536537, −2.16456689586200677242994575721, −2.12296593180616741384356879808, −1.94900829879763215761899596489, −1.75420719540305403272718199028, −1.70869271523348907545015436592, −1.47861632442151013120781440585, −1.47027173487448859332596641769, −1.23348914030851825508734487266, −0.932072389055913516288814293645, −0.862673607781792889840048745643, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.862673607781792889840048745643, 0.932072389055913516288814293645, 1.23348914030851825508734487266, 1.47027173487448859332596641769, 1.47861632442151013120781440585, 1.70869271523348907545015436592, 1.75420719540305403272718199028, 1.94900829879763215761899596489, 2.12296593180616741384356879808, 2.16456689586200677242994575721, 2.35468996523677705203841536537, 2.58181485422625611206236865058, 2.60399065431517269471434369553, 2.82851255802725079238259674345, 2.89510699653770513205567736260, 2.97982993207549499072750346163, 3.10807836138298163644508610785, 3.52883635910927260172430903801, 3.54302431425833283025794108926, 3.61671572173132762865312158428, 3.71647038354970751297885216811, 3.73024449667682994690939444576, 3.77904499124625568549472568998, 3.87890453550281960566577088392, 4.00311384923013249688251675871

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

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