Properties

Degree 32
Conductor $ 2^{32} $
Sign $1$
Motivic weight 36
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.77e5·2-s + 1.28e10·4-s + 5.81e12·5-s + 4.39e15·8-s + 7.33e17·9-s + 1.03e18·10-s + 1.94e20·13-s − 5.00e21·16-s + 4.18e22·17-s + 1.29e23·18-s + 7.50e22·20-s − 6.89e25·25-s + 3.44e25·26-s − 9.53e25·29-s − 2.93e25·32-s + 7.40e27·34-s + 9.45e27·36-s − 3.08e28·37-s + 2.55e28·40-s + 9.91e28·41-s + 4.26e30·45-s + 1.55e31·49-s − 1.22e31·50-s + 2.51e30·52-s + 2.09e31·53-s − 1.68e31·58-s + 2.16e32·61-s + ⋯
L(s)  = 1  + 0.676·2-s + 0.187·4-s + 1.52·5-s + 0.244·8-s + 4.88·9-s + 1.03·10-s + 1.73·13-s − 1.06·16-s + 2.97·17-s + 3.30·18-s + 0.286·20-s − 4.73·25-s + 1.16·26-s − 0.452·29-s − 0.0237·32-s + 2.00·34-s + 0.916·36-s − 1.82·37-s + 0.372·40-s + 0.925·41-s + 7.44·45-s + 5.86·49-s − 3.20·50-s + 0.324·52-s + 1.92·53-s − 0.306·58-s + 1.58·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(37-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+18)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(36\)
character  :  induced by $\chi_{4} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{32} ,\ ( \ : [18]^{16} ),\ 1 )\)
\(L(\frac{37}{2})\)  \(\approx\)  \(299.4127068\)
\(L(\frac12)\)  \(\approx\)  \(299.4127068\)
\(L(19)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 32. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 - 44307 p^{2} T + 289222435 p^{6} T^{2} - 1316118718197 p^{12} T^{3} + 6204095959318261 p^{20} T^{4} - 1884293613492485376 p^{30} T^{5} + 74831372726040739840 p^{42} T^{6} - \)\(16\!\cdots\!56\)\( p^{56} T^{7} + \)\(50\!\cdots\!36\)\( p^{72} T^{8} - \)\(16\!\cdots\!56\)\( p^{92} T^{9} + 74831372726040739840 p^{114} T^{10} - 1884293613492485376 p^{138} T^{11} + 6204095959318261 p^{164} T^{12} - 1316118718197 p^{192} T^{13} + 289222435 p^{222} T^{14} - 44307 p^{254} T^{15} + p^{288} T^{16} \)
good3 \( 1 - 244388283989073328 p T^{2} + \)\(43\!\cdots\!40\)\( p^{6} T^{4} - \)\(54\!\cdots\!00\)\( p^{11} T^{6} + \)\(55\!\cdots\!60\)\( p^{16} T^{8} - \)\(72\!\cdots\!72\)\( p^{29} T^{10} + \)\(84\!\cdots\!76\)\( p^{42} T^{12} - \)\(89\!\cdots\!60\)\( p^{55} T^{14} + \)\(10\!\cdots\!50\)\( p^{72} T^{16} - \)\(89\!\cdots\!60\)\( p^{127} T^{18} + \)\(84\!\cdots\!76\)\( p^{186} T^{20} - \)\(72\!\cdots\!72\)\( p^{245} T^{22} + \)\(55\!\cdots\!60\)\( p^{304} T^{24} - \)\(54\!\cdots\!00\)\( p^{371} T^{26} + \)\(43\!\cdots\!40\)\( p^{438} T^{28} - 244388283989073328 p^{505} T^{30} + p^{576} T^{32} \)
5 \( ( 1 - 581608953936 p T + \)\(37\!\cdots\!28\)\( p^{3} T^{2} - \)\(13\!\cdots\!92\)\( p^{4} T^{3} + \)\(92\!\cdots\!44\)\( p^{7} T^{4} + \)\(24\!\cdots\!52\)\( p^{12} T^{5} + \)\(59\!\cdots\!48\)\( p^{16} T^{6} + \)\(11\!\cdots\!76\)\( p^{21} T^{7} - \)\(10\!\cdots\!26\)\( p^{27} T^{8} + \)\(11\!\cdots\!76\)\( p^{57} T^{9} + \)\(59\!\cdots\!48\)\( p^{88} T^{10} + \)\(24\!\cdots\!52\)\( p^{120} T^{11} + \)\(92\!\cdots\!44\)\( p^{151} T^{12} - \)\(13\!\cdots\!92\)\( p^{184} T^{13} + \)\(37\!\cdots\!28\)\( p^{219} T^{14} - 581608953936 p^{253} T^{15} + p^{288} T^{16} )^{2} \)
7 \( 1 - \)\(15\!\cdots\!84\)\( T^{2} + \)\(26\!\cdots\!20\)\( p^{2} T^{4} - \)\(46\!\cdots\!80\)\( p^{5} T^{6} + \)\(96\!\cdots\!80\)\( p^{9} T^{8} - \)\(23\!\cdots\!84\)\( p^{14} T^{10} + \)\(51\!\cdots\!68\)\( p^{19} T^{12} - \)\(97\!\cdots\!40\)\( p^{24} T^{14} + \)\(23\!\cdots\!30\)\( p^{30} T^{16} - \)\(97\!\cdots\!40\)\( p^{96} T^{18} + \)\(51\!\cdots\!68\)\( p^{163} T^{20} - \)\(23\!\cdots\!84\)\( p^{230} T^{22} + \)\(96\!\cdots\!80\)\( p^{297} T^{24} - \)\(46\!\cdots\!80\)\( p^{365} T^{26} + \)\(26\!\cdots\!20\)\( p^{434} T^{28} - \)\(15\!\cdots\!84\)\( p^{504} T^{30} + p^{576} T^{32} \)
11 \( 1 - \)\(20\!\cdots\!16\)\( T^{2} + \)\(16\!\cdots\!60\)\( p^{3} T^{4} - \)\(10\!\cdots\!60\)\( p^{4} T^{6} + \)\(41\!\cdots\!60\)\( p^{7} T^{8} - \)\(12\!\cdots\!28\)\( p^{11} T^{10} + \)\(34\!\cdots\!68\)\( p^{15} T^{12} - \)\(69\!\cdots\!20\)\( p^{21} T^{14} + \)\(12\!\cdots\!70\)\( p^{27} T^{16} - \)\(69\!\cdots\!20\)\( p^{93} T^{18} + \)\(34\!\cdots\!68\)\( p^{159} T^{20} - \)\(12\!\cdots\!28\)\( p^{227} T^{22} + \)\(41\!\cdots\!60\)\( p^{295} T^{24} - \)\(10\!\cdots\!60\)\( p^{364} T^{26} + \)\(16\!\cdots\!60\)\( p^{435} T^{28} - \)\(20\!\cdots\!16\)\( p^{504} T^{30} + p^{576} T^{32} \)
13 \( ( 1 - 97306007082829587088 T + \)\(49\!\cdots\!60\)\( T^{2} - \)\(31\!\cdots\!44\)\( p T^{3} + \)\(78\!\cdots\!24\)\( p^{2} T^{4} - \)\(42\!\cdots\!08\)\( p^{3} T^{5} + \)\(67\!\cdots\!60\)\( p^{5} T^{6} - \)\(24\!\cdots\!72\)\( p^{7} T^{7} + \)\(33\!\cdots\!82\)\( p^{9} T^{8} - \)\(24\!\cdots\!72\)\( p^{43} T^{9} + \)\(67\!\cdots\!60\)\( p^{77} T^{10} - \)\(42\!\cdots\!08\)\( p^{111} T^{11} + \)\(78\!\cdots\!24\)\( p^{146} T^{12} - \)\(31\!\cdots\!44\)\( p^{181} T^{13} + \)\(49\!\cdots\!60\)\( p^{216} T^{14} - 97306007082829587088 p^{252} T^{15} + p^{288} T^{16} )^{2} \)
17 \( ( 1 - \)\(12\!\cdots\!64\)\( p T + \)\(39\!\cdots\!00\)\( p^{2} T^{2} - \)\(30\!\cdots\!44\)\( p^{3} T^{3} + \)\(61\!\cdots\!96\)\( p^{4} T^{4} - \)\(19\!\cdots\!04\)\( p^{6} T^{5} + \)\(20\!\cdots\!40\)\( p^{8} T^{6} - \)\(47\!\cdots\!36\)\( p^{10} T^{7} + \)\(51\!\cdots\!26\)\( p^{12} T^{8} - \)\(47\!\cdots\!36\)\( p^{46} T^{9} + \)\(20\!\cdots\!40\)\( p^{80} T^{10} - \)\(19\!\cdots\!04\)\( p^{114} T^{11} + \)\(61\!\cdots\!96\)\( p^{148} T^{12} - \)\(30\!\cdots\!44\)\( p^{183} T^{13} + \)\(39\!\cdots\!00\)\( p^{218} T^{14} - \)\(12\!\cdots\!64\)\( p^{253} T^{15} + p^{288} T^{16} )^{2} \)
19 \( 1 - \)\(96\!\cdots\!36\)\( T^{2} + \)\(44\!\cdots\!60\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{8} - \)\(13\!\cdots\!28\)\( p^{2} T^{10} + \)\(51\!\cdots\!68\)\( p^{4} T^{12} - \)\(17\!\cdots\!20\)\( p^{6} T^{14} + \)\(54\!\cdots\!70\)\( p^{8} T^{16} - \)\(17\!\cdots\!20\)\( p^{78} T^{18} + \)\(51\!\cdots\!68\)\( p^{148} T^{20} - \)\(13\!\cdots\!28\)\( p^{218} T^{22} + \)\(28\!\cdots\!60\)\( p^{288} T^{24} - \)\(13\!\cdots\!60\)\( p^{360} T^{26} + \)\(44\!\cdots\!60\)\( p^{432} T^{28} - \)\(96\!\cdots\!36\)\( p^{504} T^{30} + p^{576} T^{32} \)
23 \( 1 - \)\(10\!\cdots\!24\)\( T^{2} + \)\(51\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{6} + \)\(75\!\cdots\!20\)\( p^{2} T^{8} - \)\(26\!\cdots\!76\)\( p^{4} T^{10} + \)\(77\!\cdots\!76\)\( p^{6} T^{12} - \)\(18\!\cdots\!40\)\( p^{8} T^{14} + \)\(76\!\cdots\!70\)\( p^{12} T^{16} - \)\(18\!\cdots\!40\)\( p^{80} T^{18} + \)\(77\!\cdots\!76\)\( p^{150} T^{20} - \)\(26\!\cdots\!76\)\( p^{220} T^{22} + \)\(75\!\cdots\!20\)\( p^{290} T^{24} - \)\(16\!\cdots\!60\)\( p^{360} T^{26} + \)\(51\!\cdots\!60\)\( p^{432} T^{28} - \)\(10\!\cdots\!24\)\( p^{504} T^{30} + p^{576} T^{32} \)
29 \( ( 1 + \)\(47\!\cdots\!76\)\( T + \)\(81\!\cdots\!00\)\( p T^{2} + \)\(25\!\cdots\!40\)\( p^{2} T^{3} + \)\(10\!\cdots\!60\)\( p^{3} T^{4} - \)\(82\!\cdots\!36\)\( p^{4} T^{5} + \)\(87\!\cdots\!36\)\( p^{5} T^{6} - \)\(11\!\cdots\!40\)\( p^{6} T^{7} + \)\(52\!\cdots\!30\)\( p^{7} T^{8} - \)\(11\!\cdots\!40\)\( p^{42} T^{9} + \)\(87\!\cdots\!36\)\( p^{77} T^{10} - \)\(82\!\cdots\!36\)\( p^{112} T^{11} + \)\(10\!\cdots\!60\)\( p^{147} T^{12} + \)\(25\!\cdots\!40\)\( p^{182} T^{13} + \)\(81\!\cdots\!00\)\( p^{217} T^{14} + \)\(47\!\cdots\!76\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
31 \( 1 - \)\(49\!\cdots\!36\)\( T^{2} + \)\(13\!\cdots\!60\)\( p^{2} T^{4} - \)\(23\!\cdots\!60\)\( p^{4} T^{6} + \)\(30\!\cdots\!60\)\( p^{6} T^{8} - \)\(30\!\cdots\!88\)\( p^{8} T^{10} + \)\(25\!\cdots\!28\)\( p^{10} T^{12} - \)\(17\!\cdots\!20\)\( p^{12} T^{14} + \)\(94\!\cdots\!70\)\( p^{14} T^{16} - \)\(17\!\cdots\!20\)\( p^{84} T^{18} + \)\(25\!\cdots\!28\)\( p^{154} T^{20} - \)\(30\!\cdots\!88\)\( p^{224} T^{22} + \)\(30\!\cdots\!60\)\( p^{294} T^{24} - \)\(23\!\cdots\!60\)\( p^{364} T^{26} + \)\(13\!\cdots\!60\)\( p^{434} T^{28} - \)\(49\!\cdots\!36\)\( p^{504} T^{30} + p^{576} T^{32} \)
37 \( ( 1 + \)\(15\!\cdots\!92\)\( T + \)\(13\!\cdots\!40\)\( T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!56\)\( T^{4} + \)\(91\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!36\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} + \)\(32\!\cdots\!36\)\( p^{36} T^{9} + \)\(38\!\cdots\!60\)\( p^{72} T^{10} + \)\(91\!\cdots\!24\)\( p^{108} T^{11} + \)\(90\!\cdots\!56\)\( p^{144} T^{12} + \)\(17\!\cdots\!08\)\( p^{180} T^{13} + \)\(13\!\cdots\!40\)\( p^{216} T^{14} + \)\(15\!\cdots\!92\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
41 \( ( 1 - \)\(49\!\cdots\!56\)\( T + \)\(54\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(37\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!08\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!90\)\( T^{8} - \)\(56\!\cdots\!40\)\( p^{36} T^{9} + \)\(21\!\cdots\!08\)\( p^{72} T^{10} - \)\(37\!\cdots\!88\)\( p^{108} T^{11} + \)\(13\!\cdots\!20\)\( p^{144} T^{12} - \)\(19\!\cdots\!20\)\( p^{180} T^{13} + \)\(54\!\cdots\!20\)\( p^{216} T^{14} - \)\(49\!\cdots\!56\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
43 \( 1 - \)\(46\!\cdots\!24\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(29\!\cdots\!56\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{12} - \)\(21\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!50\)\( T^{16} - \)\(21\!\cdots\!20\)\( p^{72} T^{18} + \)\(27\!\cdots\!04\)\( p^{144} T^{20} - \)\(29\!\cdots\!56\)\( p^{216} T^{22} + \)\(27\!\cdots\!00\)\( p^{288} T^{24} - \)\(20\!\cdots\!00\)\( p^{360} T^{26} + \)\(11\!\cdots\!20\)\( p^{432} T^{28} - \)\(46\!\cdots\!24\)\( p^{504} T^{30} + p^{576} T^{32} \)
47 \( 1 - \)\(20\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(60\!\cdots\!00\)\( T^{8} - \)\(21\!\cdots\!96\)\( T^{10} + \)\(57\!\cdots\!84\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!20\)\( p^{72} T^{18} + \)\(57\!\cdots\!84\)\( p^{144} T^{20} - \)\(21\!\cdots\!96\)\( p^{216} T^{22} + \)\(60\!\cdots\!00\)\( p^{288} T^{24} - \)\(13\!\cdots\!00\)\( p^{360} T^{26} + \)\(20\!\cdots\!20\)\( p^{432} T^{28} - \)\(20\!\cdots\!64\)\( p^{504} T^{30} + p^{576} T^{32} \)
53 \( ( 1 - \)\(19\!\cdots\!76\)\( p T + \)\(82\!\cdots\!00\)\( T^{2} - \)\(68\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!36\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(65\!\cdots\!00\)\( T^{6} - \)\(36\!\cdots\!24\)\( T^{7} + \)\(94\!\cdots\!86\)\( T^{8} - \)\(36\!\cdots\!24\)\( p^{36} T^{9} + \)\(65\!\cdots\!00\)\( p^{72} T^{10} - \)\(20\!\cdots\!56\)\( p^{108} T^{11} + \)\(30\!\cdots\!36\)\( p^{144} T^{12} - \)\(68\!\cdots\!12\)\( p^{180} T^{13} + \)\(82\!\cdots\!00\)\( p^{216} T^{14} - \)\(19\!\cdots\!76\)\( p^{253} T^{15} + p^{288} T^{16} )^{2} \)
59 \( 1 - \)\(53\!\cdots\!36\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!20\)\( T^{8} - \)\(41\!\cdots\!48\)\( T^{10} + \)\(37\!\cdots\!88\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!50\)\( T^{16} - \)\(27\!\cdots\!40\)\( p^{72} T^{18} + \)\(37\!\cdots\!88\)\( p^{144} T^{20} - \)\(41\!\cdots\!48\)\( p^{216} T^{22} + \)\(38\!\cdots\!20\)\( p^{288} T^{24} - \)\(27\!\cdots\!00\)\( p^{360} T^{26} + \)\(14\!\cdots\!20\)\( p^{432} T^{28} - \)\(53\!\cdots\!36\)\( p^{504} T^{30} + p^{576} T^{32} \)
61 \( ( 1 - \)\(10\!\cdots\!16\)\( T + \)\(99\!\cdots\!20\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - \)\(55\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!90\)\( T^{8} - \)\(15\!\cdots\!40\)\( p^{36} T^{9} + \)\(14\!\cdots\!88\)\( p^{72} T^{10} - \)\(55\!\cdots\!88\)\( p^{108} T^{11} + \)\(47\!\cdots\!20\)\( p^{144} T^{12} - \)\(11\!\cdots\!20\)\( p^{180} T^{13} + \)\(99\!\cdots\!20\)\( p^{216} T^{14} - \)\(10\!\cdots\!16\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
67 \( 1 - \)\(44\!\cdots\!04\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!40\)\( T^{8} - \)\(22\!\cdots\!56\)\( T^{10} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{14} + \)\(77\!\cdots\!10\)\( T^{16} - \)\(13\!\cdots\!80\)\( p^{72} T^{18} + \)\(18\!\cdots\!84\)\( p^{144} T^{20} - \)\(22\!\cdots\!56\)\( p^{216} T^{22} + \)\(22\!\cdots\!40\)\( p^{288} T^{24} - \)\(17\!\cdots\!80\)\( p^{360} T^{26} + \)\(10\!\cdots\!40\)\( p^{432} T^{28} - \)\(44\!\cdots\!04\)\( p^{504} T^{30} + p^{576} T^{32} \)
71 \( 1 - \)\(33\!\cdots\!16\)\( T^{2} + \)\(57\!\cdots\!20\)\( T^{4} - \)\(70\!\cdots\!00\)\( T^{6} + \)\(66\!\cdots\!20\)\( T^{8} - \)\(50\!\cdots\!48\)\( T^{10} + \)\(65\!\cdots\!08\)\( p^{2} T^{12} - \)\(71\!\cdots\!40\)\( p^{4} T^{14} + \)\(67\!\cdots\!50\)\( p^{6} T^{16} - \)\(71\!\cdots\!40\)\( p^{76} T^{18} + \)\(65\!\cdots\!08\)\( p^{146} T^{20} - \)\(50\!\cdots\!48\)\( p^{216} T^{22} + \)\(66\!\cdots\!20\)\( p^{288} T^{24} - \)\(70\!\cdots\!00\)\( p^{360} T^{26} + \)\(57\!\cdots\!20\)\( p^{432} T^{28} - \)\(33\!\cdots\!16\)\( p^{504} T^{30} + p^{576} T^{32} \)
73 \( ( 1 + \)\(67\!\cdots\!32\)\( T + \)\(51\!\cdots\!60\)\( T^{2} + \)\(28\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!96\)\( T^{4} + \)\(75\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!96\)\( T^{7} + \)\(48\!\cdots\!66\)\( T^{8} + \)\(12\!\cdots\!96\)\( p^{36} T^{9} + \)\(33\!\cdots\!80\)\( p^{72} T^{10} + \)\(75\!\cdots\!24\)\( p^{108} T^{11} + \)\(17\!\cdots\!96\)\( p^{144} T^{12} + \)\(28\!\cdots\!68\)\( p^{180} T^{13} + \)\(51\!\cdots\!60\)\( p^{216} T^{14} + \)\(67\!\cdots\!32\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
79 \( 1 - \)\(10\!\cdots\!76\)\( T^{2} + \)\(62\!\cdots\!60\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(90\!\cdots\!60\)\( T^{8} - \)\(26\!\cdots\!08\)\( T^{10} + \)\(71\!\cdots\!48\)\( T^{12} - \)\(21\!\cdots\!80\)\( p T^{14} + \)\(36\!\cdots\!70\)\( T^{16} - \)\(21\!\cdots\!80\)\( p^{73} T^{18} + \)\(71\!\cdots\!48\)\( p^{144} T^{20} - \)\(26\!\cdots\!08\)\( p^{216} T^{22} + \)\(90\!\cdots\!60\)\( p^{288} T^{24} - \)\(26\!\cdots\!60\)\( p^{360} T^{26} + \)\(62\!\cdots\!60\)\( p^{432} T^{28} - \)\(10\!\cdots\!76\)\( p^{504} T^{30} + p^{576} T^{32} \)
83 \( 1 - \)\(73\!\cdots\!24\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{10} + \)\(30\!\cdots\!44\)\( T^{12} - \)\(43\!\cdots\!40\)\( T^{14} + \)\(57\!\cdots\!70\)\( T^{16} - \)\(43\!\cdots\!40\)\( p^{72} T^{18} + \)\(30\!\cdots\!44\)\( p^{144} T^{20} - \)\(20\!\cdots\!96\)\( p^{216} T^{22} + \)\(12\!\cdots\!40\)\( p^{288} T^{24} - \)\(66\!\cdots\!60\)\( p^{360} T^{26} + \)\(27\!\cdots\!00\)\( p^{432} T^{28} - \)\(73\!\cdots\!24\)\( p^{504} T^{30} + p^{576} T^{32} \)
89 \( ( 1 - \)\(23\!\cdots\!44\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!00\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!24\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!50\)\( T^{8} - \)\(11\!\cdots\!20\)\( p^{36} T^{9} + \)\(10\!\cdots\!24\)\( p^{72} T^{10} - \)\(55\!\cdots\!56\)\( p^{108} T^{11} + \)\(42\!\cdots\!00\)\( p^{144} T^{12} - \)\(16\!\cdots\!00\)\( p^{180} T^{13} + \)\(10\!\cdots\!20\)\( p^{216} T^{14} - \)\(23\!\cdots\!44\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
97 \( ( 1 - \)\(22\!\cdots\!48\)\( T + \)\(14\!\cdots\!40\)\( T^{2} - \)\(61\!\cdots\!12\)\( T^{3} + \)\(83\!\cdots\!76\)\( T^{4} + \)\(19\!\cdots\!04\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!16\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} + \)\(16\!\cdots\!16\)\( p^{36} T^{9} + \)\(26\!\cdots\!00\)\( p^{72} T^{10} + \)\(19\!\cdots\!04\)\( p^{108} T^{11} + \)\(83\!\cdots\!76\)\( p^{144} T^{12} - \)\(61\!\cdots\!12\)\( p^{180} T^{13} + \)\(14\!\cdots\!40\)\( p^{216} T^{14} - \)\(22\!\cdots\!48\)\( p^{252} T^{15} + p^{288} T^{16} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.58291757322873207206874775784, −2.52851057964172917842280157606, −2.49806010945054053509721309488, −2.33038252803916961404326009595, −2.30507337557090686801408920116, −2.20204161769867009707526607515, −1.94451972714920004295378240031, −1.89237082299382099286779909451, −1.82228881647261449157466752290, −1.76197976849507879358279702937, −1.66782833515984445829227209120, −1.66740501937398034412756259438, −1.33491121188158257291189230602, −1.19712069706608386776815285995, −1.18036922855491536329562800152, −1.16519279409216299872931555142, −1.15150936161162853110955585664, −1.06896055379920513725414306427, −0.72137168172718499739154869840, −0.68385309521737848678694755155, −0.58027044328429848954052299218, −0.48137984665521758223199382163, −0.36659457287038227066141878196, −0.29639372819466899118334388165, −0.13776279888503047230612645353, 0.13776279888503047230612645353, 0.29639372819466899118334388165, 0.36659457287038227066141878196, 0.48137984665521758223199382163, 0.58027044328429848954052299218, 0.68385309521737848678694755155, 0.72137168172718499739154869840, 1.06896055379920513725414306427, 1.15150936161162853110955585664, 1.16519279409216299872931555142, 1.18036922855491536329562800152, 1.19712069706608386776815285995, 1.33491121188158257291189230602, 1.66740501937398034412756259438, 1.66782833515984445829227209120, 1.76197976849507879358279702937, 1.82228881647261449157466752290, 1.89237082299382099286779909451, 1.94451972714920004295378240031, 2.20204161769867009707526607515, 2.30507337557090686801408920116, 2.33038252803916961404326009595, 2.49806010945054053509721309488, 2.52851057964172917842280157606, 2.58291757322873207206874775784

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

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