Properties

Degree 96
Conductor $ 3^{144} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 6·3-s + 15·4-s + 6·5-s + 36·6-s − 6·7-s − 27·8-s − 9·9-s − 36·10-s + 57·11-s − 90·12-s − 6·13-s + 36·14-s − 36·15-s + 57·16-s − 207·17-s + 54·18-s − 3·19-s + 90·20-s + 36·21-s − 342·22-s + 402·23-s + 162·24-s − 93·25-s + 36·26-s + 501·27-s − 90·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 15/8·4-s + 0.536·5-s + 2.44·6-s − 0.323·7-s − 1.19·8-s − 1/3·9-s − 1.13·10-s + 1.56·11-s − 2.16·12-s − 0.128·13-s + 0.687·14-s − 0.619·15-s + 0.890·16-s − 2.95·17-s + 0.707·18-s − 0.0362·19-s + 1.00·20-s + 0.374·21-s − 3.31·22-s + 3.64·23-s + 1.37·24-s − 0.743·25-s + 0.271·26-s + 3.57·27-s − 0.607·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{144}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{144}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{48} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(96\)
\( N \)  =  \(3^{144}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{27} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(96,\ 3^{144} ,\ ( \ : [3/2]^{48} ),\ 1 )$
$L(2)$  $\approx$  $2.95617$
$L(\frac12)$  $\approx$  $2.95617$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 96. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 95.
$p$$F_p(T)$
bad3 \( 1 + 2 p T + 5 p^{2} T^{2} - 59 p T^{3} - 245 p^{2} T^{4} - 115 p^{4} T^{5} + 976 p^{2} T^{6} + 2731 p^{5} T^{7} + 19084 p^{4} T^{8} - 10325 p^{4} T^{9} - 148021 p^{6} T^{10} - 598831 p^{6} T^{11} + 1456744 p^{6} T^{12} + 574066 p^{9} T^{13} + 4401880 p^{9} T^{14} - 6202085 p^{9} T^{15} - 7669748 p^{11} T^{16} - 3436025 p^{13} T^{17} + 13112023 p^{12} T^{18} + 53965820 p^{14} T^{19} + 5278532 p^{16} T^{20} - 31225340 p^{16} T^{21} - 41749115 p^{18} T^{22} - 1025683 p^{21} T^{23} + 128275240 p^{20} T^{24} - 1025683 p^{24} T^{25} - 41749115 p^{24} T^{26} - 31225340 p^{25} T^{27} + 5278532 p^{28} T^{28} + 53965820 p^{29} T^{29} + 13112023 p^{30} T^{30} - 3436025 p^{34} T^{31} - 7669748 p^{35} T^{32} - 6202085 p^{36} T^{33} + 4401880 p^{39} T^{34} + 574066 p^{42} T^{35} + 1456744 p^{42} T^{36} - 598831 p^{45} T^{37} - 148021 p^{48} T^{38} - 10325 p^{49} T^{39} + 19084 p^{52} T^{40} + 2731 p^{56} T^{41} + 976 p^{56} T^{42} - 115 p^{61} T^{43} - 245 p^{62} T^{44} - 59 p^{64} T^{45} + 5 p^{68} T^{46} + 2 p^{70} T^{47} + p^{72} T^{48} \)
good2 \( 1 + 3 p T + 21 T^{2} + 63 T^{3} + 21 p^{3} T^{4} + 537 T^{5} + 709 p T^{6} + 5355 T^{7} + 10593 p T^{8} + 21519 p^{2} T^{9} + 5409 p^{6} T^{10} + 532287 p T^{11} + 379893 p^{3} T^{12} + 8415531 T^{13} + 25855263 T^{14} + 9559431 p^{3} T^{15} + 232601427 T^{16} + 818763975 T^{17} + 1314151619 p T^{18} + 8191032423 T^{19} + 11991895359 p T^{20} + 66654259137 T^{21} + 182988883995 T^{22} + 256410458499 p T^{23} + 1474496834597 T^{24} + 2012114428749 p T^{25} + 12073142225529 T^{26} + 9092271645009 p^{2} T^{27} + 99715752705771 T^{28} + 272130697887345 T^{29} + 745766101178463 T^{30} + 931335629537493 p T^{31} + 1067185500779025 p^{2} T^{32} + 335231443417545 p^{5} T^{33} + 424050111197415 p^{6} T^{34} + 532044570212847 p^{7} T^{35} + 88688261905595 p^{11} T^{36} + 1793179876029693 p^{8} T^{37} + 3683424379143297 p^{8} T^{38} + 4084861974833367 p^{9} T^{39} + 52110345788889 p^{16} T^{40} - 1179749736325785 p^{12} T^{41} - 7058191587035509 p^{12} T^{42} - 8979058827105021 p^{13} T^{43} - 16261400269717659 p^{14} T^{44} - 34689337128478845 p^{15} T^{45} - 48278286287516313 p^{16} T^{46} - 32180700011527845 p^{18} T^{47} - 106038377789220459 p^{18} T^{48} - 32180700011527845 p^{21} T^{49} - 48278286287516313 p^{22} T^{50} - 34689337128478845 p^{24} T^{51} - 16261400269717659 p^{26} T^{52} - 8979058827105021 p^{28} T^{53} - 7058191587035509 p^{30} T^{54} - 1179749736325785 p^{33} T^{55} + 52110345788889 p^{40} T^{56} + 4084861974833367 p^{36} T^{57} + 3683424379143297 p^{38} T^{58} + 1793179876029693 p^{41} T^{59} + 88688261905595 p^{47} T^{60} + 532044570212847 p^{46} T^{61} + 424050111197415 p^{48} T^{62} + 335231443417545 p^{50} T^{63} + 1067185500779025 p^{50} T^{64} + 931335629537493 p^{52} T^{65} + 745766101178463 p^{54} T^{66} + 272130697887345 p^{57} T^{67} + 99715752705771 p^{60} T^{68} + 9092271645009 p^{65} T^{69} + 12073142225529 p^{66} T^{70} + 2012114428749 p^{70} T^{71} + 1474496834597 p^{72} T^{72} + 256410458499 p^{76} T^{73} + 182988883995 p^{78} T^{74} + 66654259137 p^{81} T^{75} + 11991895359 p^{85} T^{76} + 8191032423 p^{87} T^{77} + 1314151619 p^{91} T^{78} + 818763975 p^{93} T^{79} + 232601427 p^{96} T^{80} + 9559431 p^{102} T^{81} + 25855263 p^{102} T^{82} + 8415531 p^{105} T^{83} + 379893 p^{111} T^{84} + 532287 p^{112} T^{85} + 5409 p^{120} T^{86} + 21519 p^{119} T^{87} + 10593 p^{121} T^{88} + 5355 p^{123} T^{89} + 709 p^{127} T^{90} + 537 p^{129} T^{91} + 21 p^{135} T^{92} + 63 p^{135} T^{93} + 21 p^{138} T^{94} + 3 p^{142} T^{95} + p^{144} T^{96} \)
5 \( 1 - 6 T + 129 T^{2} + 1143 T^{3} - 21648 T^{4} + 108579 T^{5} - 1528213 T^{6} - 18987228 T^{7} - 68437161 p T^{8} - 389172546 T^{9} + 46145282184 T^{10} - 1024779866094 T^{11} + 2377546179321 p T^{12} + 124731059904432 T^{13} - 1267020688405716 T^{14} + 4535842062540393 p T^{15} + 38601136618335342 T^{16} - 2418705521255322 p^{2} T^{17} - 1773759072521851828 p T^{18} - 95897648595023515212 T^{19} + \)\(20\!\cdots\!01\)\( T^{20} - \)\(78\!\cdots\!97\)\( T^{21} + \)\(78\!\cdots\!14\)\( T^{22} + \)\(64\!\cdots\!33\)\( p T^{23} - \)\(89\!\cdots\!01\)\( T^{24} + \)\(66\!\cdots\!86\)\( T^{25} + \)\(65\!\cdots\!21\)\( p T^{26} + \)\(50\!\cdots\!74\)\( T^{27} + \)\(48\!\cdots\!73\)\( T^{28} - \)\(26\!\cdots\!98\)\( p^{2} T^{29} + \)\(22\!\cdots\!08\)\( T^{30} - \)\(17\!\cdots\!29\)\( T^{31} - \)\(64\!\cdots\!81\)\( T^{32} + \)\(38\!\cdots\!74\)\( p T^{33} - \)\(40\!\cdots\!09\)\( T^{34} - \)\(40\!\cdots\!29\)\( T^{35} - \)\(71\!\cdots\!59\)\( T^{36} - \)\(88\!\cdots\!62\)\( T^{37} + \)\(66\!\cdots\!94\)\( T^{38} - \)\(18\!\cdots\!69\)\( T^{39} + \)\(80\!\cdots\!16\)\( T^{40} - \)\(17\!\cdots\!23\)\( T^{41} - \)\(84\!\cdots\!78\)\( T^{42} + \)\(12\!\cdots\!68\)\( T^{43} - \)\(58\!\cdots\!87\)\( T^{44} - \)\(25\!\cdots\!07\)\( p T^{45} + \)\(34\!\cdots\!84\)\( T^{46} - \)\(10\!\cdots\!52\)\( T^{47} - \)\(29\!\cdots\!06\)\( T^{48} - \)\(10\!\cdots\!52\)\( p^{3} T^{49} + \)\(34\!\cdots\!84\)\( p^{6} T^{50} - \)\(25\!\cdots\!07\)\( p^{10} T^{51} - \)\(58\!\cdots\!87\)\( p^{12} T^{52} + \)\(12\!\cdots\!68\)\( p^{15} T^{53} - \)\(84\!\cdots\!78\)\( p^{18} T^{54} - \)\(17\!\cdots\!23\)\( p^{21} T^{55} + \)\(80\!\cdots\!16\)\( p^{24} T^{56} - \)\(18\!\cdots\!69\)\( p^{27} T^{57} + \)\(66\!\cdots\!94\)\( p^{30} T^{58} - \)\(88\!\cdots\!62\)\( p^{33} T^{59} - \)\(71\!\cdots\!59\)\( p^{36} T^{60} - \)\(40\!\cdots\!29\)\( p^{39} T^{61} - \)\(40\!\cdots\!09\)\( p^{42} T^{62} + \)\(38\!\cdots\!74\)\( p^{46} T^{63} - \)\(64\!\cdots\!81\)\( p^{48} T^{64} - \)\(17\!\cdots\!29\)\( p^{51} T^{65} + \)\(22\!\cdots\!08\)\( p^{54} T^{66} - \)\(26\!\cdots\!98\)\( p^{59} T^{67} + \)\(48\!\cdots\!73\)\( p^{60} T^{68} + \)\(50\!\cdots\!74\)\( p^{63} T^{69} + \)\(65\!\cdots\!21\)\( p^{67} T^{70} + \)\(66\!\cdots\!86\)\( p^{69} T^{71} - \)\(89\!\cdots\!01\)\( p^{72} T^{72} + \)\(64\!\cdots\!33\)\( p^{76} T^{73} + \)\(78\!\cdots\!14\)\( p^{78} T^{74} - \)\(78\!\cdots\!97\)\( p^{81} T^{75} + \)\(20\!\cdots\!01\)\( p^{84} T^{76} - 95897648595023515212 p^{87} T^{77} - 1773759072521851828 p^{91} T^{78} - 2418705521255322 p^{95} T^{79} + 38601136618335342 p^{96} T^{80} + 4535842062540393 p^{100} T^{81} - 1267020688405716 p^{102} T^{82} + 124731059904432 p^{105} T^{83} + 2377546179321 p^{109} T^{84} - 1024779866094 p^{111} T^{85} + 46145282184 p^{114} T^{86} - 389172546 p^{117} T^{87} - 68437161 p^{121} T^{88} - 18987228 p^{123} T^{89} - 1528213 p^{126} T^{90} + 108579 p^{129} T^{91} - 21648 p^{132} T^{92} + 1143 p^{135} T^{93} + 129 p^{138} T^{94} - 6 p^{141} T^{95} + p^{144} T^{96} \)
7 \( 1 + 6T + 318T^{2} + 1.15e3T^{3} - 3.46e4T^{4} - 4.84e6T^{5} - 5.81e7T^{6} - 2.57e9T^{7} - 2.35e10T^{8} + 2.89e11T^{9} + 1.38e13T^{10} + 2.85e14T^{11} + 9.54e15T^{12} + 4.47e16T^{13} - 7.90e17T^{14} - 3.73e19T^{15} - 1.07e21T^{16} - 2.80e22T^{17} + 1.27e23T^{18} + 4.19e24T^{19} + 1.59e26T^{20} + 4.03e27T^{21} + 4.57e28T^{22} - 7.20e29T^{23} - 1.69e31T^{24} - 5.97e32T^{25} - 1.27e34T^{26} - 4.12e34T^{27} + 2.57e36T^{28} + 5.95e37T^{29} + 1.94e39T^{30} + 2.23e40T^{31} - 9.08e40T^{32} - 8.24e42T^{33} - 1.91e44T^{34} - 4.95e45T^{35} - 2.97e46T^{36} + 7.07e47T^{37} + 2.47e49T^{38} + 6.14e50T^{39} + 9.44e51T^{40} + 1.33e52T^{41} - 2.42e54T^{42} - 6.47e55T^{43} - 1.33e57T^{44}+O(T^{45}) \)
11 \( 1 - 57T + 183T^{2} + 1.75e4T^{3} - 1.52e6T^{4} + 3.43e8T^{5} - 1.17e10T^{6} - 1.96e11T^{7} + 1.32e13T^{8} - 7.31e14T^{9} + 5.72e16T^{10} - 8.14e17T^{11} - 6.60e19T^{12} + 2.53e21T^{13} - 1.07e23T^{14} + 5.07e24T^{15} + 5.61e25T^{16} - 9.82e27T^{17} + 2.28e29T^{18} - 8.94e30T^{19} + 2.74e32T^{20} + 1.46e34T^{21} - 7.72e35T^{22} + 9.52e36T^{23} - 6.20e38T^{24} + 1.29e40T^{25} + 1.26e42T^{26} - 3.51e43T^{27} + 3.77e44T^{28} - 6.42e46T^{29} + 1.06e48T^{30} + 6.85e49T^{31} - 1.18e51T^{32} + 7.67e52T^{33} - 6.85e54T^{34} + 6.81e55T^{35} + 3.68e57T^{36}+O(T^{37}) \)
13 \( 1 + 6T - 5.48e3T^{2} - 1.70e5T^{3} + 1.29e7T^{4} + 6.87e7T^{5} - 1.62e10T^{6} + 1.32e12T^{7} + 9.32e13T^{8} - 2.27e15T^{9} - 1.50e16T^{10} + 3.08e18T^{11} - 6.09e20T^{12} - 3.82e22T^{13} + 3.14e23T^{14} + 3.89e24T^{15} - 1.41e27T^{16} + 2.15e29T^{17} + 1.44e31T^{18} + 4.27e31T^{19} - 1.31e33T^{20} + 7.71e35T^{21} - 4.03e37T^{22} - 4.30e39T^{23} - 7.69e40T^{24} - 2.34e42T^{25} - 3.03e44T^{26} + 5.58e45T^{27} + 1.06e48T^{28} + 2.01e49T^{29} + 9.95e50T^{30} + 1.14e53T^{31} + 1.72e54T^{32} - 1.91e56T^{33} - 4.83e57T^{34}+O(T^{35}) \)
17 \( 1 + 207T - 4.09e4T^{2} - 1.13e7T^{3} + 6.96e8T^{4} + 3.06e11T^{5} - 4.50e12T^{6} - 5.47e15T^{7} - 5.25e16T^{8} + 7.34e19T^{9} + 1.83e21T^{10} - 7.92e23T^{11} - 2.83e25T^{12} + 7.15e27T^{13} + 3.11e29T^{14} - 5.57e31T^{15} - 2.70e33T^{16} + 3.81e35T^{17} + 1.91e37T^{18} - 2.33e39T^{19} - 1.10e41T^{20} + 1.28e43T^{21} + 4.92e44T^{22} - 6.48e46T^{23} - 1.31e48T^{24} + 2.96e50T^{25} - 3.34e51T^{26} - 1.20e54T^{27} + 7.93e55T^{28} + 4.16e57T^{29} - 7.12e59T^{30} - 1.02e61T^{31}+O(T^{32}) \)
19 \( 1 + 3T - 8.77e4T^{2} - 2.08e6T^{3} + 3.87e9T^{4} + 1.64e11T^{5} - 1.13e14T^{6} - 6.81e15T^{7} + 2.46e18T^{8} + 1.90e20T^{9} - 4.23e22T^{10} - 4.02e24T^{11} + 5.96e26T^{12} + 6.81e28T^{13} - 7.06e30T^{14} - 9.61e32T^{15} + 7.20e34T^{16} + 1.15e37T^{17} - 6.46e38T^{18} - 1.20e41T^{19} + 5.26e42T^{20} + 1.10e45T^{21} - 4.08e46T^{22} - 8.87e48T^{23} + 3.24e50T^{24} + 6.21e52T^{25} - 2.80e54T^{26} - 3.71e56T^{27} + 2.60e58T^{28} + 1.78e60T^{29} - 2.45e62T^{30}+O(T^{31}) \)
23 \( 1 - 402T + 1.56e5T^{2} - 4.70e7T^{3} + 1.25e10T^{4} - 3.04e12T^{5} + 6.79e14T^{6} - 1.42e17T^{7} + 2.81e19T^{8} - 5.31e21T^{9} + 9.61e23T^{10} - 1.67e26T^{11} + 2.82e28T^{12} - 4.60e30T^{13} + 7.31e32T^{14} - 1.13e35T^{15} + 1.70e37T^{16} - 2.52e39T^{17} + 3.65e41T^{18} - 5.18e43T^{19} + 7.22e45T^{20} - 9.88e47T^{21} + 1.33e50T^{22} - 1.76e52T^{23} + 2.30e54T^{24} - 2.97e56T^{25} + 3.79e58T^{26} - 4.76e60T^{27} + 5.92e62T^{28}+O(T^{29}) \)
29 \( 1 - 480T + 2.22e5T^{2} - 7.76e7T^{3} + 2.55e10T^{4} - 7.40e12T^{5} + 2.04e15T^{6} - 5.15e17T^{7} + 1.25e20T^{8} - 2.83e22T^{9} + 6.18e24T^{10} - 1.27e27T^{11} + 2.52e29T^{12} - 4.72e31T^{13} + 8.50e33T^{14} - 1.43e36T^{15} + 2.31e38T^{16} - 3.42e40T^{17} + 4.65e42T^{18} - 5.29e44T^{19} + 4.26e46T^{20} + 1.36e48T^{21} - 1.52e51T^{22} + 4.56e53T^{23} - 1.00e56T^{24} + 1.94e58T^{25} - 3.35e60T^{26}+O(T^{27}) \)
31 \( 1 + 60T - 1.48e5T^{2} - 1.22e7T^{3} + 1.01e10T^{4} + 8.79e11T^{5} - 4.03e14T^{6} - 2.80e16T^{7} + 9.57e18T^{8} + 7.09e19T^{9} - 3.82e22T^{10} + 3.03e25T^{11} - 8.45e27T^{12} - 1.24e30T^{13} + 4.56e32T^{14} + 1.68e34T^{15} - 1.30e37T^{16} + 3.70e38T^{17} + 9.06e40T^{18} - 2.84e43T^{19} + 1.11e46T^{20} + 1.30e48T^{21} - 5.76e50T^{22} - 5.05e52T^{23} + 1.32e55T^{24} + 1.45e57T^{25}+O(T^{26}) \)
37 \( 1 + 3T - 6.65e5T^{2} + 2.82e7T^{3} + 2.26e11T^{4} - 1.94e13T^{5} - 5.18e16T^{6} + 6.69e18T^{7} + 8.88e21T^{8} - 1.54e24T^{9} - 1.20e27T^{10} + 2.70e29T^{11} + 1.32e32T^{12} - 3.78e34T^{13} - 1.18e37T^{14} + 4.40e39T^{15} + 8.63e41T^{16} - 4.34e44T^{17} - 4.77e46T^{18} + 3.68e49T^{19} + 1.54e51T^{20} - 2.69e54T^{21} + 4.21e55T^{22} + 1.70e59T^{23} - 1.15e61T^{24}+O(T^{25}) \)
41 \( 1 + 1.73e3T + 1.74e6T^{2} + 1.23e9T^{3} + 6.95e11T^{4} + 3.24e14T^{5} + 1.32e17T^{6} + 4.80e19T^{7} + 1.60e22T^{8} + 4.96e24T^{9} + 1.45e27T^{10} + 4.06e29T^{11} + 1.11e32T^{12} + 2.98e34T^{13} + 7.86e36T^{14} + 1.96e39T^{15} + 4.43e41T^{16} + 8.26e43T^{17} + 8.95e45T^{18} - 1.84e48T^{19} - 1.66e51T^{20} - 7.28e53T^{21} - 2.51e56T^{22} - 7.67e58T^{23} - 2.16e61T^{24}+O(T^{25}) \)
43 \( 1 - 507T + 3.11e5T^{2} - 1.74e8T^{3} + 8.72e10T^{4} - 4.40e13T^{5} + 1.92e16T^{6} - 8.29e18T^{7} + 3.46e21T^{8} - 1.38e24T^{9} + 5.41e26T^{10} - 2.01e29T^{11} + 7.34e31T^{12} - 2.62e34T^{13} + 9.08e36T^{14} - 3.08e39T^{15} + 1.01e42T^{16} - 3.28e44T^{17} + 1.04e47T^{18} - 3.22e49T^{19} + 9.78e51T^{20} - 2.88e54T^{21} + 8.35e56T^{22} - 2.36e59T^{23}+O(T^{24}) \)
47 \( 1 - 984T + 5.16e5T^{2} - 2.62e8T^{3} + 1.34e11T^{4} - 6.13e13T^{5} + 2.73e16T^{6} - 1.09e19T^{7} + 3.92e21T^{8} - 1.48e24T^{9} + 5.65e26T^{10} - 1.89e29T^{11} + 6.12e31T^{12} - 1.91e34T^{13} + 5.87e36T^{14} - 1.98e39T^{15} + 5.87e41T^{16} - 1.37e44T^{17} + 3.65e46T^{18} - 1.00e49T^{19} + 2.21e51T^{20} - 2.23e53T^{21} - 2.38e56T^{22} + 1.53e59T^{23}+O(T^{24}) \)
53 \( 1 - 2.73e3T + 7.98e6T^{2} - 1.48e10T^{3} + 2.67e13T^{4} - 3.92e16T^{5} + 5.45e19T^{6} - 6.69e22T^{7} + 7.80e25T^{8} - 8.35e28T^{9} + 8.53e31T^{10} - 8.16e34T^{11} + 7.49e37T^{12} - 6.52e40T^{13} + 5.46e43T^{14} - 4.38e46T^{15} + 3.39e49T^{16} - 2.53e52T^{17} + 1.83e55T^{18} - 1.28e58T^{19} + 8.70e60T^{20} - 5.73e63T^{21} + 3.68e66T^{22}+O(T^{23}) \)
59 \( 1 - 2.23e3T + 3.50e6T^{2} - 4.35e9T^{3} + 4.69e12T^{4} - 4.49e15T^{5} + 3.93e18T^{6} - 3.18e21T^{7} + 2.41e24T^{8} - 1.72e27T^{9} + 1.17e30T^{10} - 7.55e32T^{11} + 4.66e35T^{12} - 2.76e38T^{13} + 1.57e41T^{14} - 8.59e43T^{15} + 4.52e46T^{16} - 2.29e49T^{17} + 1.12e52T^{18} - 5.28e54T^{19} + 2.38e57T^{20} - 1.03e60T^{21} + 4.27e62T^{22}+O(T^{23}) \)
61 \( 1 - 48T - 1.72e5T^{2} - 7.72e7T^{3} - 3.38e10T^{4} + 4.83e12T^{5} + 3.04e16T^{6} + 8.65e18T^{7} - 1.28e21T^{8} - 3.31e24T^{9} - 1.81e27T^{10} - 5.40e29T^{11} + 4.14e32T^{12} + 2.45e35T^{13} + 7.77e37T^{14} + 5.61e38T^{15} - 2.83e43T^{16} - 1.51e46T^{17} - 2.71e48T^{18} + 3.44e50T^{19} + 1.41e54T^{20} + 7.77e56T^{21}+O(T^{22}) \)
67 \( 1 + 681T + 5.84e5T^{2} + 4.00e8T^{3} + 3.69e11T^{4} + 1.80e14T^{5} + 1.13e17T^{6} + 5.74e19T^{7} + 3.16e22T^{8} + 7.32e24T^{9} + 5.96e27T^{10} - 1.02e30T^{11} - 1.61e33T^{12} - 1.78e36T^{13} - 7.28e38T^{14} - 8.46e41T^{15} - 4.06e44T^{16} - 1.97e47T^{17} - 8.10e49T^{18} - 6.20e52T^{19} - 7.30e54T^{20} - 7.11e57T^{21}+O(T^{22}) \)
71 \( 1 + 3.10e3T - 1.53e5T^{2} - 1.03e10T^{3} - 7.09e12T^{4} + 1.66e16T^{5} + 1.97e19T^{6} - 1.64e22T^{7} - 3.07e25T^{8} + 9.54e27T^{9} + 3.40e31T^{10} - 7.24e32T^{11} - 2.91e37T^{12} - 5.31e39T^{13} + 2.01e43T^{14} + 7.11e45T^{15} - 1.14e49T^{16} - 5.96e51T^{17} + 5.38e54T^{18} + 3.82e57T^{19} - 2.04e60T^{20} - 1.99e63T^{21}+O(T^{22}) \)
73 \( 1 + 219T - 5.83e6T^{2} - 7.38e8T^{3} + 1.73e13T^{4} + 7.10e14T^{5} - 3.48e19T^{6} + 1.35e21T^{7} + 5.28e25T^{8} - 5.99e27T^{9} - 6.42e31T^{10} + 1.18e34T^{11} + 6.50e37T^{12} - 1.63e40T^{13} - 5.62e43T^{14} + 1.77e46T^{15} + 4.22e49T^{16} - 1.59e52T^{17} - 2.79e55T^{18} + 1.21e58T^{19} + 1.65e61T^{20} - 8.07e63T^{21}+O(T^{22}) \)
79 \( 1 - 2.80e3T + 2.09e6T^{2} + 1.45e9T^{3} - 3.89e12T^{4} + 2.50e15T^{5} + 1.54e18T^{6} - 3.83e21T^{7} + 2.32e24T^{8} + 3.48e26T^{9} - 1.71e30T^{10} + 1.41e33T^{11} - 4.21e35T^{12} - 3.29e38T^{13} + 7.72e41T^{14} - 8.15e44T^{15} + 3.45e47T^{16} + 2.47e50T^{17} - 5.13e53T^{18} + 3.34e56T^{19} + 2.04e58T^{20}+O(T^{21}) \)
83 \( 1 - 3.46e3T + 4.59e6T^{2} - 3.11e9T^{3} + 1.62e12T^{4} - 7.83e13T^{5} - 2.10e18T^{6} + 1.94e21T^{7} + 3.08e23T^{8} - 1.05e27T^{9} + 5.99e29T^{10} - 5.51e32T^{11} - 8.44e34T^{12} + 1.09e39T^{13} - 8.24e41T^{14} - 7.73e43T^{15} + 2.11e47T^{16} - 1.05e50T^{17} + 2.23e53T^{18} - 4.49e55T^{19} - 2.35e59T^{20}+O(T^{21}) \)
89 \( 1 + 5.20e3T + 5.78e6T^{2} - 1.70e10T^{3} - 4.62e13T^{4} + 3.03e14T^{5} + 1.15e20T^{6} + 9.16e22T^{7} - 1.45e26T^{8} - 2.33e29T^{9} + 8.12e31T^{10} + 3.35e35T^{11} + 5.28e37T^{12} - 3.34e41T^{13} - 1.84e44T^{14} + 2.26e47T^{15} + 2.44e50T^{16} - 6.00e52T^{17} - 1.98e56T^{18} - 8.31e58T^{19} + 7.68e61T^{20}+O(T^{21}) \)
97 \( 1 + 3.38e3T + 7.02e6T^{2} + 9.02e9T^{3} + 1.06e13T^{4} + 1.03e16T^{5} + 1.01e19T^{6} + 2.53e21T^{7} - 6.18e24T^{8} - 1.97e28T^{9} - 2.35e31T^{10} - 3.06e34T^{11} - 2.54e37T^{12} - 2.54e40T^{13} - 3.14e42T^{14} + 9.93e45T^{15} + 3.83e49T^{16} + 3.70e52T^{17} + 5.14e55T^{18} + 3.33e58T^{19}+O(T^{20}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{96} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.96703175399403924478302794791, −2.75330951754792471959701340071, −2.59276614497512186505977617562, −2.58614205665582351900876399237, −2.54649121263162921782917100208, −2.40021000513703018189083034113, −2.39867116015528794661633291394, −2.30508093884605609361262026822, −2.26729969812163849923429875686, −2.20208299318965194795403470130, −2.08666146997398087738577047962, −2.07480092707585363972836243542, −1.98787838818610936603492684719, −1.77408341863338498162756458293, −1.77130883060899360815301648393, −1.55119488685298694706377128376, −1.42069217992721931366796728410, −1.39200601833495428808173821772, −1.02285008517443497630368547100, −0.941049926643177377366575243940, −0.825891913404535097636267097145, −0.71653981401384729909148117373, −0.63009003674638723270792936967, −0.57748247998975939620741274177, −0.34709593077116122194215721108, 0.34709593077116122194215721108, 0.57748247998975939620741274177, 0.63009003674638723270792936967, 0.71653981401384729909148117373, 0.825891913404535097636267097145, 0.941049926643177377366575243940, 1.02285008517443497630368547100, 1.39200601833495428808173821772, 1.42069217992721931366796728410, 1.55119488685298694706377128376, 1.77130883060899360815301648393, 1.77408341863338498162756458293, 1.98787838818610936603492684719, 2.07480092707585363972836243542, 2.08666146997398087738577047962, 2.20208299318965194795403470130, 2.26729969812163849923429875686, 2.30508093884605609361262026822, 2.39867116015528794661633291394, 2.40021000513703018189083034113, 2.54649121263162921782917100208, 2.58614205665582351900876399237, 2.59276614497512186505977617562, 2.75330951754792471959701340071, 2.96703175399403924478302794791

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

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