Properties

Degree 72
Conductor $ 3^{36} \cdot 5^{36} \cdot 7^{36} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 18·4-s − 58·7-s + 7·9-s − 72·12-s − 100·13-s + 195·16-s + 50·19-s − 232·21-s + 45·25-s + 8·27-s + 1.04e3·28-s − 82·31-s − 126·36-s − 26·37-s − 400·39-s − 204·43-s + 780·48-s + 1.49e3·49-s + 1.80e3·52-s + 200·57-s − 324·61-s − 406·63-s − 1.55e3·64-s − 142·67-s + 386·73-s + 180·75-s + ⋯
L(s)  = 1  + 4/3·3-s − 9/2·4-s − 8.28·7-s + 7/9·9-s − 6·12-s − 7.69·13-s + 12.1·16-s + 2.63·19-s − 11.0·21-s + 9/5·25-s + 8/27·27-s + 37.2·28-s − 2.64·31-s − 7/2·36-s − 0.702·37-s − 10.2·39-s − 4.74·43-s + 65/4·48-s + 30.4·49-s + 34.6·52-s + 3.50·57-s − 5.31·61-s − 6.44·63-s − 24.2·64-s − 2.11·67-s + 5.28·73-s + 12/5·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{36} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{36} \cdot 7^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(72\)
\( N \)  =  \(3^{36} \cdot 5^{36} \cdot 7^{36}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((72,\ 3^{36} \cdot 5^{36} \cdot 7^{36} ,\ ( \ : [1]^{36} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00165075\)
\(L(\frac12)\)  \(\approx\)  \(0.00165075\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 72. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 71.
$p$$F_p(T)$
bad3 \( 1 - 4 T + p^{2} T^{2} - 16 T^{3} - 22 T^{4} + 76 p^{2} T^{5} - 1283 T^{6} + 272 T^{7} + 4942 T^{8} - 14596 p T^{9} + 30769 p^{2} T^{10} - 128176 T^{11} - 278915 p T^{12} + 3416380 T^{13} - 16710038 T^{14} + 6856852 p^{2} T^{15} + 3788744 p^{3} T^{16} - 6289300 p^{4} T^{17} + 17004050 p^{4} T^{18} - 6289300 p^{6} T^{19} + 3788744 p^{7} T^{20} + 6856852 p^{8} T^{21} - 16710038 p^{8} T^{22} + 3416380 p^{10} T^{23} - 278915 p^{13} T^{24} - 128176 p^{14} T^{25} + 30769 p^{18} T^{26} - 14596 p^{19} T^{27} + 4942 p^{20} T^{28} + 272 p^{22} T^{29} - 1283 p^{24} T^{30} + 76 p^{28} T^{31} - 22 p^{28} T^{32} - 16 p^{30} T^{33} + p^{34} T^{34} - 4 p^{34} T^{35} + p^{36} T^{36} \)
5 \( ( 1 - p T^{2} + p^{2} T^{4} )^{9} \)
7 \( ( 1 + 29 T + 516 T^{2} + 6997 T^{3} + 81846 T^{4} + 837791 T^{5} + 7779887 T^{6} + 65786348 T^{7} + 73483926 p T^{8} + 10844146 p^{3} T^{9} + 73483926 p^{3} T^{10} + 65786348 p^{4} T^{11} + 7779887 p^{6} T^{12} + 837791 p^{8} T^{13} + 81846 p^{10} T^{14} + 6997 p^{12} T^{15} + 516 p^{14} T^{16} + 29 p^{16} T^{17} + p^{18} T^{18} )^{2} \)
good2 \( 1 + 9 p T^{2} + 129 T^{4} + 183 p T^{6} - 257 p T^{8} - 3313 p T^{10} - 1971 T^{12} + 131001 p T^{14} + 1020279 p T^{16} + 3434233 p T^{18} + 3073977 T^{20} - 27945015 p T^{22} - 120303967 T^{24} + 283803891 p^{2} T^{26} + 1468283337 p^{3} T^{28} + 6580731125 p^{3} T^{30} + 3453009269 p^{5} T^{32} - 359979793 p^{6} T^{34} - 48655109319 p^{4} T^{36} - 359979793 p^{10} T^{38} + 3453009269 p^{13} T^{40} + 6580731125 p^{15} T^{42} + 1468283337 p^{19} T^{44} + 283803891 p^{22} T^{46} - 120303967 p^{24} T^{48} - 27945015 p^{29} T^{50} + 3073977 p^{32} T^{52} + 3434233 p^{37} T^{54} + 1020279 p^{41} T^{56} + 131001 p^{45} T^{58} - 1971 p^{48} T^{60} - 3313 p^{53} T^{62} - 257 p^{57} T^{64} + 183 p^{61} T^{66} + 129 p^{64} T^{68} + 9 p^{69} T^{70} + p^{72} T^{72} \)
11 \( 1 + 788 T^{2} + 278026 T^{4} + 5131804 p T^{6} + 6957278853 T^{8} + 508994176716 T^{10} + 21349626073812 T^{12} - 1518091294375584 T^{14} - 1343544847946842290 T^{16} - \)\(36\!\cdots\!48\)\( T^{18} - \)\(51\!\cdots\!04\)\( T^{20} - \)\(40\!\cdots\!84\)\( T^{22} - \)\(27\!\cdots\!34\)\( T^{24} - \)\(36\!\cdots\!20\)\( T^{26} - \)\(25\!\cdots\!88\)\( T^{28} + \)\(59\!\cdots\!04\)\( T^{30} + \)\(15\!\cdots\!39\)\( T^{32} + \)\(15\!\cdots\!56\)\( T^{34} + \)\(12\!\cdots\!30\)\( T^{36} + \)\(15\!\cdots\!56\)\( p^{4} T^{38} + \)\(15\!\cdots\!39\)\( p^{8} T^{40} + \)\(59\!\cdots\!04\)\( p^{12} T^{42} - \)\(25\!\cdots\!88\)\( p^{16} T^{44} - \)\(36\!\cdots\!20\)\( p^{20} T^{46} - \)\(27\!\cdots\!34\)\( p^{24} T^{48} - \)\(40\!\cdots\!84\)\( p^{28} T^{50} - \)\(51\!\cdots\!04\)\( p^{32} T^{52} - \)\(36\!\cdots\!48\)\( p^{36} T^{54} - 1343544847946842290 p^{40} T^{56} - 1518091294375584 p^{44} T^{58} + 21349626073812 p^{48} T^{60} + 508994176716 p^{52} T^{62} + 6957278853 p^{56} T^{64} + 5131804 p^{61} T^{66} + 278026 p^{64} T^{68} + 788 p^{68} T^{70} + p^{72} T^{72} \)
13 \( ( 1 + 25 T + 1095 T^{2} + 22948 T^{3} + 564575 T^{4} + 9997031 T^{5} + 183902851 T^{6} + 2780640376 T^{7} + 42276086786 T^{8} + 549551140388 T^{9} + 42276086786 p^{2} T^{10} + 2780640376 p^{4} T^{11} + 183902851 p^{6} T^{12} + 9997031 p^{8} T^{13} + 564575 p^{10} T^{14} + 22948 p^{12} T^{15} + 1095 p^{14} T^{16} + 25 p^{16} T^{17} + p^{18} T^{18} )^{4} \)
17 \( 1 + 2122 T^{2} + 2177527 T^{4} + 1517394490 T^{6} + 848361547073 T^{8} + 412443398864484 T^{10} + 179742134685083008 T^{12} + 71738156809829153164 T^{14} + \)\(15\!\cdots\!26\)\( p T^{16} + \)\(95\!\cdots\!16\)\( T^{18} + \)\(32\!\cdots\!10\)\( T^{20} + \)\(10\!\cdots\!20\)\( T^{22} + \)\(35\!\cdots\!94\)\( T^{24} + \)\(11\!\cdots\!56\)\( T^{26} + \)\(36\!\cdots\!68\)\( T^{28} + \)\(11\!\cdots\!36\)\( T^{30} + \)\(21\!\cdots\!07\)\( p T^{32} + \)\(63\!\cdots\!10\)\( p T^{34} + \)\(31\!\cdots\!81\)\( T^{36} + \)\(63\!\cdots\!10\)\( p^{5} T^{38} + \)\(21\!\cdots\!07\)\( p^{9} T^{40} + \)\(11\!\cdots\!36\)\( p^{12} T^{42} + \)\(36\!\cdots\!68\)\( p^{16} T^{44} + \)\(11\!\cdots\!56\)\( p^{20} T^{46} + \)\(35\!\cdots\!94\)\( p^{24} T^{48} + \)\(10\!\cdots\!20\)\( p^{28} T^{50} + \)\(32\!\cdots\!10\)\( p^{32} T^{52} + \)\(95\!\cdots\!16\)\( p^{36} T^{54} + \)\(15\!\cdots\!26\)\( p^{41} T^{56} + 71738156809829153164 p^{44} T^{58} + 179742134685083008 p^{48} T^{60} + 412443398864484 p^{52} T^{62} + 848361547073 p^{56} T^{64} + 1517394490 p^{60} T^{66} + 2177527 p^{64} T^{68} + 2122 p^{68} T^{70} + p^{72} T^{72} \)
19 \( ( 1 - 25 T - 1200 T^{2} + 28279 T^{3} + 727998 T^{4} - 13341725 T^{5} - 336272232 T^{6} + 3996483707 T^{7} + 127959530986 T^{8} - 1043089831645 T^{9} - 44989163301640 T^{10} + 212357710438651 T^{11} + 18123889516039151 T^{12} - 165504901739946 T^{13} - 7258586647908305128 T^{14} - 15789080398634467314 T^{15} + \)\(23\!\cdots\!40\)\( T^{16} + \)\(30\!\cdots\!14\)\( T^{17} - \)\(74\!\cdots\!00\)\( T^{18} + \)\(30\!\cdots\!14\)\( p^{2} T^{19} + \)\(23\!\cdots\!40\)\( p^{4} T^{20} - 15789080398634467314 p^{6} T^{21} - 7258586647908305128 p^{8} T^{22} - 165504901739946 p^{10} T^{23} + 18123889516039151 p^{12} T^{24} + 212357710438651 p^{14} T^{25} - 44989163301640 p^{16} T^{26} - 1043089831645 p^{18} T^{27} + 127959530986 p^{20} T^{28} + 3996483707 p^{22} T^{29} - 336272232 p^{24} T^{30} - 13341725 p^{26} T^{31} + 727998 p^{28} T^{32} + 28279 p^{30} T^{33} - 1200 p^{32} T^{34} - 25 p^{34} T^{35} + p^{36} T^{36} )^{2} \)
23 \( 1 + 5456 T^{2} + 15641132 T^{4} + 30816565844 T^{6} + 46397715523648 T^{8} + 56434832036216860 T^{10} + 57338507001711805874 T^{12} + \)\(49\!\cdots\!88\)\( T^{14} + \)\(37\!\cdots\!40\)\( T^{16} + \)\(24\!\cdots\!12\)\( T^{18} + \)\(14\!\cdots\!20\)\( T^{20} + \)\(77\!\cdots\!68\)\( T^{22} + \)\(39\!\cdots\!83\)\( T^{24} + \)\(20\!\cdots\!96\)\( T^{26} + \)\(11\!\cdots\!52\)\( T^{28} + \)\(31\!\cdots\!12\)\( p T^{30} + \)\(45\!\cdots\!32\)\( T^{32} + \)\(27\!\cdots\!48\)\( T^{34} + \)\(15\!\cdots\!36\)\( T^{36} + \)\(27\!\cdots\!48\)\( p^{4} T^{38} + \)\(45\!\cdots\!32\)\( p^{8} T^{40} + \)\(31\!\cdots\!12\)\( p^{13} T^{42} + \)\(11\!\cdots\!52\)\( p^{16} T^{44} + \)\(20\!\cdots\!96\)\( p^{20} T^{46} + \)\(39\!\cdots\!83\)\( p^{24} T^{48} + \)\(77\!\cdots\!68\)\( p^{28} T^{50} + \)\(14\!\cdots\!20\)\( p^{32} T^{52} + \)\(24\!\cdots\!12\)\( p^{36} T^{54} + \)\(37\!\cdots\!40\)\( p^{40} T^{56} + \)\(49\!\cdots\!88\)\( p^{44} T^{58} + 57338507001711805874 p^{48} T^{60} + 56434832036216860 p^{52} T^{62} + 46397715523648 p^{56} T^{64} + 30816565844 p^{60} T^{66} + 15641132 p^{64} T^{68} + 5456 p^{68} T^{70} + p^{72} T^{72} \)
29 \( ( 1 - 8702 T^{2} + 37465263 T^{4} - 105916890148 T^{6} + 220655071810143 T^{8} - 361326921340752698 T^{10} + \)\(48\!\cdots\!87\)\( T^{12} - \)\(55\!\cdots\!32\)\( T^{14} + \)\(55\!\cdots\!82\)\( T^{16} - \)\(49\!\cdots\!12\)\( T^{18} + \)\(55\!\cdots\!82\)\( p^{4} T^{20} - \)\(55\!\cdots\!32\)\( p^{8} T^{22} + \)\(48\!\cdots\!87\)\( p^{12} T^{24} - 361326921340752698 p^{16} T^{26} + 220655071810143 p^{20} T^{28} - 105916890148 p^{24} T^{30} + 37465263 p^{28} T^{32} - 8702 p^{32} T^{34} + p^{36} T^{36} )^{2} \)
31 \( ( 1 + 41 T - 2445 T^{2} - 43060 T^{3} + 5237514 T^{4} - 16938260 T^{5} - 6098172023 T^{6} + 95852721615 T^{7} + 2623669646472 T^{8} - 109999816850699 T^{9} + 1267701269242851 T^{10} + 14361146547326364 T^{11} - 3281854883381845163 T^{12} + 33876102645099343219 T^{13} + \)\(32\!\cdots\!82\)\( T^{14} - \)\(14\!\cdots\!93\)\( T^{15} - \)\(39\!\cdots\!30\)\( T^{16} + \)\(85\!\cdots\!85\)\( T^{17} + \)\(52\!\cdots\!22\)\( T^{18} + \)\(85\!\cdots\!85\)\( p^{2} T^{19} - \)\(39\!\cdots\!30\)\( p^{4} T^{20} - \)\(14\!\cdots\!93\)\( p^{6} T^{21} + \)\(32\!\cdots\!82\)\( p^{8} T^{22} + 33876102645099343219 p^{10} T^{23} - 3281854883381845163 p^{12} T^{24} + 14361146547326364 p^{14} T^{25} + 1267701269242851 p^{16} T^{26} - 109999816850699 p^{18} T^{27} + 2623669646472 p^{20} T^{28} + 95852721615 p^{22} T^{29} - 6098172023 p^{24} T^{30} - 16938260 p^{26} T^{31} + 5237514 p^{28} T^{32} - 43060 p^{30} T^{33} - 2445 p^{32} T^{34} + 41 p^{34} T^{35} + p^{36} T^{36} )^{2} \)
37 \( ( 1 + 13 T - 6076 T^{2} - 264627 T^{3} + 437312 p T^{4} + 1275247373 T^{5} - 8886886438 T^{6} - 86602899303 p T^{7} - 59893401193908 T^{8} + 4289183820546821 T^{9} + 200947887710821612 T^{10} - 1410910717925405091 T^{11} - \)\(30\!\cdots\!57\)\( T^{12} - \)\(66\!\cdots\!34\)\( T^{13} + \)\(46\!\cdots\!84\)\( p T^{14} + \)\(13\!\cdots\!70\)\( T^{15} + \)\(27\!\cdots\!00\)\( T^{16} - \)\(92\!\cdots\!10\)\( T^{17} - \)\(70\!\cdots\!44\)\( T^{18} - \)\(92\!\cdots\!10\)\( p^{2} T^{19} + \)\(27\!\cdots\!00\)\( p^{4} T^{20} + \)\(13\!\cdots\!70\)\( p^{6} T^{21} + \)\(46\!\cdots\!84\)\( p^{9} T^{22} - \)\(66\!\cdots\!34\)\( p^{10} T^{23} - \)\(30\!\cdots\!57\)\( p^{12} T^{24} - 1410910717925405091 p^{14} T^{25} + 200947887710821612 p^{16} T^{26} + 4289183820546821 p^{18} T^{27} - 59893401193908 p^{20} T^{28} - 86602899303 p^{23} T^{29} - 8886886438 p^{24} T^{30} + 1275247373 p^{26} T^{31} + 437312 p^{29} T^{32} - 264627 p^{30} T^{33} - 6076 p^{32} T^{34} + 13 p^{34} T^{35} + p^{36} T^{36} )^{2} \)
41 \( 1 - 2.58e4T^{2} + 3.45e8T^{4} - 3.16e12T^{6} + 2.24e16T^{8} - 1.29e20T^{10} + 6.39e23T^{12} - 2.74e27T^{14} + 1.05e31T^{16} - 3.61e34T^{18} + 1.12e38T^{20} - 3.23e41T^{22} + 8.54e44T^{24} - 2.09e48T^{26} + 4.75e51T^{28} - 1.00e55T^{30} + 2.00e58T^{32} - 3.71e61T^{34}+O(T^{36}) \)
43 \( 1 + 204T + 4.61e4T^{2} + 6.52e6T^{3} + 9.13e8T^{4} + 1.03e11T^{5} + 1.13e13T^{6} + 1.09e15T^{7} + 1.02e17T^{8} + 8.76e18T^{9} + 7.28e20T^{10} + 5.69e22T^{11} + 4.32e24T^{12} + 3.12e26T^{13} + 2.19e28T^{14} + 1.48e30T^{15} + 9.82e31T^{16} + 6.27e33T^{17} + 3.91e35T^{18} + 2.37e37T^{19} + 1.40e39T^{20} + 8.11e40T^{21} + 4.59e42T^{22} + 2.53e44T^{23} + 1.37e46T^{24} + 7.30e47T^{25} + 3.80e49T^{26} + 1.93e51T^{27} + 9.71e52T^{28} + 4.76e54T^{29} + 2.30e56T^{30} + 1.08e58T^{31} + 5.07e59T^{32} + 2.31e61T^{33} + 1.04e63T^{34} + 4.59e64T^{35}+O(T^{36}) \)
47 \( 1 + 2.08e4T^{2} + 2.22e8T^{4} + 1.60e12T^{6} + 8.64e15T^{8} + 3.67e19T^{10} + 1.25e23T^{12} + 3.48e26T^{14} + 7.45e29T^{16} + 1.00e33T^{18} - 4.51e35T^{20} - 8.17e39T^{22} - 3.09e43T^{24} - 7.71e46T^{26} - 1.29e50T^{28} - 7.64e52T^{30} + 4.01e56T^{32} + 1.91e60T^{34}+O(T^{35}) \)
53 \( 1 + 2.40e4T^{2} + 2.95e8T^{4} + 2.39e12T^{6} + 1.39e16T^{8} + 6.02e19T^{10} + 1.84e23T^{12} + 3.03e26T^{14} - 5.58e29T^{16} - 6.69e33T^{18} - 3.04e37T^{20} - 9.46e40T^{22} - 2.10e44T^{24} - 2.66e47T^{26} + 2.69e50T^{28} + 2.98e54T^{30} + 1.18e58T^{32}+O(T^{34}) \)
59 \( 1 + 4.11e4T^{2} + 8.59e8T^{4} + 1.21e13T^{6} + 1.31e17T^{8} + 1.15e21T^{10} + 8.56e24T^{12} + 5.54e28T^{14} + 3.18e32T^{16} + 1.64e36T^{18} + 7.83e39T^{20} + 3.46e43T^{22} + 1.44e47T^{24} + 5.81e50T^{26} + 2.25e54T^{28} + 8.58e57T^{30} + 3.19e61T^{32}+O(T^{33}) \)
61 \( 1 + 324T + 2.31e3T^{2} - 9.45e6T^{3} - 5.10e8T^{4} + 1.67e11T^{5} + 1.22e13T^{6} - 2.26e15T^{7} - 1.80e17T^{8} + 2.59e19T^{9} + 1.99e21T^{10} - 2.62e23T^{11} - 1.76e25T^{12} + 2.40e27T^{13} + 1.28e29T^{14} - 1.98e31T^{15} - 7.64e32T^{16} + 1.48e35T^{17} + 3.49e36T^{18} - 1.01e39T^{19} - 9.16e39T^{20} + 6.19e42T^{21} - 3.00e43T^{22} - 3.43e46T^{23} + 6.43e47T^{24} + 1.70e50T^{25} - 5.72e51T^{26} - 7.58e53T^{27} + 3.84e55T^{28} + 2.96e57T^{29} - 2.16e59T^{30} - 9.84e60T^{31} + 1.06e63T^{32}+O(T^{33}) \)
67 \( 1 + 142T - 2.14e4T^{2} - 1.57e6T^{3} + 4.99e8T^{4} + 1.47e9T^{5} - 6.18e12T^{6} + 3.64e14T^{7} + 4.38e16T^{8} - 5.98e18T^{9} - 4.46e18T^{10} + 4.95e22T^{11} - 3.05e24T^{12} - 1.59e26T^{13} + 3.14e28T^{14} - 9.76e29T^{15} - 1.35e32T^{16} + 1.49e34T^{17} - 1.50e35T^{18} - 7.80e37T^{19} + 5.86e39T^{20} + 7.14e40T^{21} - 3.66e43T^{22} + 1.85e45T^{23} + 8.32e46T^{24} - 1.45e49T^{25} + 4.29e50T^{26} + 4.88e52T^{27} - 4.98e54T^{28} + 4.77e55T^{29} + 2.19e58T^{30} - 1.50e60T^{31}+O(T^{32}) \)
71 \( 1 - 1.07e5T^{2} + 5.78e9T^{4} - 2.07e14T^{6} + 5.58e18T^{8} - 1.19e23T^{10} + 2.13e27T^{12} - 3.24e31T^{14} + 4.29e35T^{16} - 5.02e39T^{18} + 5.25e43T^{20} - 4.95e47T^{22} + 4.24e51T^{24} - 3.32e55T^{26} + 2.38e59T^{28} - 1.58e63T^{30}+O(T^{32}) \)
73 \( 1 - 386T + 1.10e3T^{2} + 1.69e7T^{3} - 1.01e9T^{4} - 4.57e11T^{5} + 4.00e13T^{6} + 9.16e15T^{7} - 9.49e17T^{8} - 1.49e20T^{9} + 1.68e22T^{10} + 2.08e24T^{11} - 2.39e26T^{12} - 2.54e28T^{13} + 2.85e30T^{14} + 2.77e32T^{15} - 2.89e34T^{16} - 2.74e36T^{17} + 2.51e38T^{18} + 2.45e40T^{19} - 1.85e42T^{20} - 2.01e44T^{21} + 1.12e46T^{22} + 1.50e48T^{23} - 5.00e49T^{24} - 1.02e52T^{25} + 8.03e52T^{26} + 6.24e55T^{27} + 1.33e57T^{28} - 3.38e59T^{29} - 1.93e61T^{30}+O(T^{31}) \)
79 \( 1 - 334T - 1.81e4T^{2} + 1.60e7T^{3} - 5.71e7T^{4} - 4.52e11T^{5} + 7.25e12T^{6} + 9.65e15T^{7} - 1.53e17T^{8} - 1.68e20T^{9} + 1.69e21T^{10} + 2.50e24T^{11} - 4.04e24T^{12} - 3.24e28T^{13} - 2.63e29T^{14} + 3.75e32T^{15} + 6.97e33T^{16} - 3.90e36T^{17} - 1.16e38T^{18} + 3.69e40T^{19} + 1.54e42T^{20} - 3.19e44T^{21} - 1.76e46T^{22} + 2.53e48T^{23} + 1.79e50T^{24} - 1.84e52T^{25} - 1.66e54T^{26} + 1.22e56T^{27} + 1.41e58T^{28} - 7.27e59T^{29} - 1.11e62T^{30}+O(T^{31}) \)
83 \( 1 - 1.82e5T^{2} + 1.64e10T^{4} - 9.74e14T^{6} + 4.29e19T^{8} - 1.49e24T^{10} + 4.26e28T^{12} - 1.02e33T^{14} + 2.13e37T^{16} - 3.88e41T^{18} + 6.22e45T^{20} - 8.89e49T^{22} + 1.13e54T^{24} - 1.31e58T^{26} + 1.37e62T^{28} - 1.30e66T^{30}+O(T^{31}) \)
89 \( 1 + 9.24e4T^{2} + 4.38e9T^{4} + 1.42e14T^{6} + 3.57e18T^{8} + 7.28e22T^{10} + 1.25e27T^{12} + 1.88e31T^{14} + 2.46e35T^{16} + 2.83e39T^{18} + 2.86e43T^{20} + 2.48e47T^{22} + 1.76e51T^{24} + 8.66e54T^{26} - 1.86e57T^{28}+O(T^{30}) \)
97 \( 1 - 1.61e3T + 1.48e6T^{2} - 9.95e8T^{3} + 5.32e11T^{4} - 2.40e14T^{5} + 9.46e16T^{6} - 3.31e19T^{7} + 1.05e22T^{8} - 3.06e24T^{9} + 8.26e26T^{10} - 2.07e29T^{11} + 4.88e31T^{12} - 1.08e34T^{13} + 2.27e36T^{14} - 4.53e38T^{15} + 8.63e40T^{16} - 1.56e43T^{17} + 2.73e45T^{18} - 4.56e47T^{19} + 7.34e49T^{20} - 1.13e52T^{21} + 1.69e54T^{22} - 2.45e56T^{23} + 3.42e58T^{24} - 4.62e60T^{25} + 6.06e62T^{26} - 7.69e64T^{27} + 9.48e66T^{28} - 1.13e69T^{29}+O(T^{30}) \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{72} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.42205576158440448782603963383, −2.32180273531594746186054063297, −2.27538704337039420098665382761, −2.23299704710115587268685220481, −2.14640022061172408000359338128, −2.00892314074813253300250042229, −1.98944013063678869482187451486, −1.91323524993802085227931602854, −1.89728821755233347188819368403, −1.67662789714391363915844090031, −1.63611321449537247582520448422, −1.57689889156026864747022789022, −1.31230137948293484711659380068, −1.29963074633768867289222787532, −1.15250250344000629660683755292, −1.14517892939008724956997259590, −1.04372361639375584811936653019, −0.830408026951502604632650951768, −0.78164517532415639261366265017, −0.50973459560718989310457399594, −0.35006913497195498604336474585, −0.32336140297977028702953172164, −0.24914812453454057573550143046, −0.083608542765862247376480021915, −0.07687963423398083583005812465, 0.07687963423398083583005812465, 0.083608542765862247376480021915, 0.24914812453454057573550143046, 0.32336140297977028702953172164, 0.35006913497195498604336474585, 0.50973459560718989310457399594, 0.78164517532415639261366265017, 0.830408026951502604632650951768, 1.04372361639375584811936653019, 1.14517892939008724956997259590, 1.15250250344000629660683755292, 1.29963074633768867289222787532, 1.31230137948293484711659380068, 1.57689889156026864747022789022, 1.63611321449537247582520448422, 1.67662789714391363915844090031, 1.89728821755233347188819368403, 1.91323524993802085227931602854, 1.98944013063678869482187451486, 2.00892314074813253300250042229, 2.14640022061172408000359338128, 2.23299704710115587268685220481, 2.27538704337039420098665382761, 2.32180273531594746186054063297, 2.42205576158440448782603963383

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

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