Properties

Degree 24
Conductor $ 11^{12} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 6·3-s − 8·4-s − 18·5-s + 30·6-s − 80·7-s + 80·8-s + 140·9-s + 90·10-s − 43·11-s + 48·12-s + 250·13-s + 400·14-s + 108·15-s − 62·16-s − 1.25e3·17-s − 700·18-s − 1.02e3·19-s + 144·20-s + 480·21-s + 215·22-s + 1.68e3·23-s − 480·24-s + 1.19e3·25-s − 1.25e3·26-s − 1.51e3·27-s + 640·28-s + ⋯
L(s)  = 1  − 5/4·2-s − 2/3·3-s − 1/2·4-s − 0.719·5-s + 5/6·6-s − 1.63·7-s + 5/4·8-s + 1.72·9-s + 9/10·10-s − 0.355·11-s + 1/3·12-s + 1.47·13-s + 2.04·14-s + 0.479·15-s − 0.242·16-s − 4.32·17-s − 2.16·18-s − 2.83·19-s + 9/25·20-s + 1.08·21-s + 0.444·22-s + 3.18·23-s − 5/6·24-s + 1.91·25-s − 1.84·26-s − 2.07·27-s + 0.816·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(11^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(11^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{11} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 11^{12} ,\ ( \ : [2]^{12} ),\ 1 )$
$L(\frac{5}{2})$  $\approx$  $0.00230344$
$L(\frac12)$  $\approx$  $0.00230344$
$L(3)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \(F_p\) is a polynomial of degree 24. If $p = 11$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad11 \( 1 + 43 T + 1721 p T^{2} - 2065 p^{2} T^{3} + 85 p^{3} T^{4} - 3665602 p^{4} T^{5} - 1477606 p^{6} T^{6} - 3665602 p^{8} T^{7} + 85 p^{11} T^{8} - 2065 p^{14} T^{9} + 1721 p^{17} T^{10} + 43 p^{20} T^{11} + p^{24} T^{12} \)
good2 \( 1 + 5 T + 33 T^{2} + 125 T^{3} + 551 T^{4} + 905 p T^{5} + 4305 p T^{6} + 12735 p^{2} T^{7} + 58435 p^{2} T^{8} + 141605 p^{3} T^{9} + 544261 p^{3} T^{10} + 1103405 p^{4} T^{11} + 3982909 p^{4} T^{12} + 1103405 p^{8} T^{13} + 544261 p^{11} T^{14} + 141605 p^{15} T^{15} + 58435 p^{18} T^{16} + 12735 p^{22} T^{17} + 4305 p^{25} T^{18} + 905 p^{29} T^{19} + 551 p^{32} T^{20} + 125 p^{36} T^{21} + 33 p^{40} T^{22} + 5 p^{44} T^{23} + p^{48} T^{24} \)
3 \( 1 + 2 p T - 104 T^{2} + 46 T^{3} + 10081 T^{4} - 30398 p T^{5} - 163646 p^{2} T^{6} + 204208 p^{3} T^{7} + 233522 p^{5} T^{8} - 262802 p^{7} T^{9} + 16246 p^{8} T^{10} + 6196316 p^{8} T^{11} + 30056416 p^{8} T^{12} + 6196316 p^{12} T^{13} + 16246 p^{16} T^{14} - 262802 p^{19} T^{15} + 233522 p^{21} T^{16} + 204208 p^{23} T^{17} - 163646 p^{26} T^{18} - 30398 p^{29} T^{19} + 10081 p^{32} T^{20} + 46 p^{36} T^{21} - 104 p^{40} T^{22} + 2 p^{45} T^{23} + p^{48} T^{24} \)
5 \( 1 + 18 T - 874 T^{2} - 1798 p T^{3} + 184869 p T^{4} - 2498082 T^{5} - 756868316 T^{6} + 6088380428 T^{7} + 94508025588 p T^{8} - 885497879674 p T^{9} - 231003059512494 T^{10} + 256199910578148 T^{11} + 91821549616068396 T^{12} + 256199910578148 p^{4} T^{13} - 231003059512494 p^{8} T^{14} - 885497879674 p^{13} T^{15} + 94508025588 p^{17} T^{16} + 6088380428 p^{20} T^{17} - 756868316 p^{24} T^{18} - 2498082 p^{28} T^{19} + 184869 p^{33} T^{20} - 1798 p^{37} T^{21} - 874 p^{40} T^{22} + 18 p^{44} T^{23} + p^{48} T^{24} \)
7 \( 1 + 80 T + 6388 T^{2} + 701930 T^{3} + 49215681 T^{4} + 3279603800 T^{5} + 239232786210 T^{6} + 14513581978500 T^{7} + 843606939238690 T^{8} + 7237661951983100 p T^{9} + 54529399676115082 p^{2} T^{10} + 404074251544709930 p^{3} T^{11} + 3002897775232982024 p^{4} T^{12} + 404074251544709930 p^{7} T^{13} + 54529399676115082 p^{10} T^{14} + 7237661951983100 p^{13} T^{15} + 843606939238690 p^{16} T^{16} + 14513581978500 p^{20} T^{17} + 239232786210 p^{24} T^{18} + 3279603800 p^{28} T^{19} + 49215681 p^{32} T^{20} + 701930 p^{36} T^{21} + 6388 p^{40} T^{22} + 80 p^{44} T^{23} + p^{48} T^{24} \)
13 \( 1 - 250 T + 98548 T^{2} - 25239340 T^{3} + 4920135411 T^{4} - 1056434505070 T^{5} + 134058098615820 T^{6} - 14343324685617900 T^{7} + 158785723907559280 T^{8} + \)\(58\!\cdots\!50\)\( T^{9} - \)\(16\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!10\)\( T^{11} - \)\(73\!\cdots\!16\)\( T^{12} + \)\(38\!\cdots\!10\)\( p^{4} T^{13} - \)\(16\!\cdots\!92\)\( p^{8} T^{14} + \)\(58\!\cdots\!50\)\( p^{12} T^{15} + 158785723907559280 p^{16} T^{16} - 14343324685617900 p^{20} T^{17} + 134058098615820 p^{24} T^{18} - 1056434505070 p^{28} T^{19} + 4920135411 p^{32} T^{20} - 25239340 p^{36} T^{21} + 98548 p^{40} T^{22} - 250 p^{44} T^{23} + p^{48} T^{24} \)
17 \( 1 + 1250 T + 992428 T^{2} + 582264010 T^{3} + 283914863531 T^{4} + 119803876725030 T^{5} + 46223383217047820 T^{6} + 16729167329461506640 T^{7} + \)\(58\!\cdots\!20\)\( T^{8} + \)\(19\!\cdots\!90\)\( T^{9} + \)\(62\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(56\!\cdots\!04\)\( T^{12} + \)\(19\!\cdots\!80\)\( p^{4} T^{13} + \)\(62\!\cdots\!28\)\( p^{8} T^{14} + \)\(19\!\cdots\!90\)\( p^{12} T^{15} + \)\(58\!\cdots\!20\)\( p^{16} T^{16} + 16729167329461506640 p^{20} T^{17} + 46223383217047820 p^{24} T^{18} + 119803876725030 p^{28} T^{19} + 283914863531 p^{32} T^{20} + 582264010 p^{36} T^{21} + 992428 p^{40} T^{22} + 1250 p^{44} T^{23} + p^{48} T^{24} \)
19 \( 1 + 1025 T + 619378 T^{2} + 353530160 T^{3} + 166899801006 T^{4} + 66114359518655 T^{5} + 21592456954609665 T^{6} + 5902640524678934610 T^{7} + \)\(15\!\cdots\!25\)\( T^{8} + \)\(24\!\cdots\!45\)\( T^{9} - \)\(10\!\cdots\!67\)\( T^{10} - \)\(11\!\cdots\!95\)\( T^{11} - \)\(61\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!95\)\( p^{4} T^{13} - \)\(10\!\cdots\!67\)\( p^{8} T^{14} + \)\(24\!\cdots\!45\)\( p^{12} T^{15} + \)\(15\!\cdots\!25\)\( p^{16} T^{16} + 5902640524678934610 p^{20} T^{17} + 21592456954609665 p^{24} T^{18} + 66114359518655 p^{28} T^{19} + 166899801006 p^{32} T^{20} + 353530160 p^{36} T^{21} + 619378 p^{40} T^{22} + 1025 p^{44} T^{23} + p^{48} T^{24} \)
23 \( ( 1 - 842 T + 1237966 T^{2} - 916714810 T^{3} + 731144219855 T^{4} - 446744415652252 T^{5} + 258141193233856964 T^{6} - 446744415652252 p^{4} T^{7} + 731144219855 p^{8} T^{8} - 916714810 p^{12} T^{9} + 1237966 p^{16} T^{10} - 842 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
29 \( 1 + 2690 T + 5931748 T^{2} + 10598261380 T^{3} + 16033149600451 T^{4} + 21947778968733670 T^{5} + 27405046849774036980 T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!60\)\( T^{8} + \)\(34\!\cdots\!50\)\( T^{9} + \)\(33\!\cdots\!68\)\( T^{10} + \)\(30\!\cdots\!90\)\( T^{11} + \)\(26\!\cdots\!84\)\( T^{12} + \)\(30\!\cdots\!90\)\( p^{4} T^{13} + \)\(33\!\cdots\!68\)\( p^{8} T^{14} + \)\(34\!\cdots\!50\)\( p^{12} T^{15} + \)\(34\!\cdots\!60\)\( p^{16} T^{16} + \)\(31\!\cdots\!20\)\( p^{20} T^{17} + 27405046849774036980 p^{24} T^{18} + 21947778968733670 p^{28} T^{19} + 16033149600451 p^{32} T^{20} + 10598261380 p^{36} T^{21} + 5931748 p^{40} T^{22} + 2690 p^{44} T^{23} + p^{48} T^{24} \)
31 \( 1 + 1136 T - 230384 T^{2} - 239729514 T^{3} + 1267435488321 T^{4} + 2677537350048516 T^{5} + 1095876078080394266 T^{6} - \)\(12\!\cdots\!44\)\( T^{7} + \)\(51\!\cdots\!26\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(25\!\cdots\!46\)\( T^{10} + \)\(85\!\cdots\!26\)\( T^{11} - \)\(77\!\cdots\!44\)\( T^{12} + \)\(85\!\cdots\!26\)\( p^{4} T^{13} + \)\(25\!\cdots\!46\)\( p^{8} T^{14} + \)\(24\!\cdots\!96\)\( p^{12} T^{15} + \)\(51\!\cdots\!26\)\( p^{16} T^{16} - \)\(12\!\cdots\!44\)\( p^{20} T^{17} + 1095876078080394266 p^{24} T^{18} + 2677537350048516 p^{28} T^{19} + 1267435488321 p^{32} T^{20} - 239729514 p^{36} T^{21} - 230384 p^{40} T^{22} + 1136 p^{44} T^{23} + p^{48} T^{24} \)
37 \( 1 + 336 T + 72946 T^{2} - 4475893024 T^{3} + 1626992896581 T^{4} - 4706417114238944 T^{5} + 10320605504003112056 T^{6} - \)\(11\!\cdots\!84\)\( T^{7} + \)\(22\!\cdots\!56\)\( T^{8} - \)\(44\!\cdots\!44\)\( T^{9} + \)\(30\!\cdots\!46\)\( T^{10} - \)\(47\!\cdots\!04\)\( T^{11} + \)\(15\!\cdots\!76\)\( T^{12} - \)\(47\!\cdots\!04\)\( p^{4} T^{13} + \)\(30\!\cdots\!46\)\( p^{8} T^{14} - \)\(44\!\cdots\!44\)\( p^{12} T^{15} + \)\(22\!\cdots\!56\)\( p^{16} T^{16} - \)\(11\!\cdots\!84\)\( p^{20} T^{17} + 10320605504003112056 p^{24} T^{18} - 4706417114238944 p^{28} T^{19} + 1626992896581 p^{32} T^{20} - 4475893024 p^{36} T^{21} + 72946 p^{40} T^{22} + 336 p^{44} T^{23} + p^{48} T^{24} \)
41 \( 1 + 4550 T + 14401928 T^{2} + 23922427770 T^{3} + 36931794819271 T^{4} + 40163345738311410 T^{5} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!20\)\( T^{7} + \)\(86\!\cdots\!60\)\( T^{8} + \)\(14\!\cdots\!70\)\( T^{9} + \)\(24\!\cdots\!68\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{11} + \)\(43\!\cdots\!24\)\( T^{12} + \)\(25\!\cdots\!80\)\( p^{4} T^{13} + \)\(24\!\cdots\!68\)\( p^{8} T^{14} + \)\(14\!\cdots\!70\)\( p^{12} T^{15} + \)\(86\!\cdots\!60\)\( p^{16} T^{16} + \)\(30\!\cdots\!20\)\( p^{20} T^{17} + \)\(12\!\cdots\!20\)\( p^{24} T^{18} + 40163345738311410 p^{28} T^{19} + 36931794819271 p^{32} T^{20} + 23922427770 p^{36} T^{21} + 14401928 p^{40} T^{22} + 4550 p^{44} T^{23} + p^{48} T^{24} \)
43 \( 1 - 23356307 T^{2} + 286833712371781 T^{4} - \)\(23\!\cdots\!15\)\( T^{6} + \)\(14\!\cdots\!15\)\( T^{8} - \)\(69\!\cdots\!82\)\( T^{10} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(69\!\cdots\!82\)\( p^{8} T^{14} + \)\(14\!\cdots\!15\)\( p^{16} T^{16} - \)\(23\!\cdots\!15\)\( p^{24} T^{18} + 286833712371781 p^{32} T^{20} - 23356307 p^{40} T^{22} + p^{48} T^{24} \)
47 \( 1 - 24 T - 10068774 T^{2} - 8140867214 T^{3} + 53779050778611 T^{4} + 233185473449483776 T^{5} - 81521520124710716194 T^{6} - \)\(19\!\cdots\!44\)\( T^{7} - \)\(12\!\cdots\!74\)\( T^{8} + \)\(86\!\cdots\!36\)\( T^{9} + \)\(22\!\cdots\!96\)\( T^{10} - \)\(14\!\cdots\!34\)\( T^{11} - \)\(15\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!34\)\( p^{4} T^{13} + \)\(22\!\cdots\!96\)\( p^{8} T^{14} + \)\(86\!\cdots\!36\)\( p^{12} T^{15} - \)\(12\!\cdots\!74\)\( p^{16} T^{16} - \)\(19\!\cdots\!44\)\( p^{20} T^{17} - 81521520124710716194 p^{24} T^{18} + 233185473449483776 p^{28} T^{19} + 53779050778611 p^{32} T^{20} - 8140867214 p^{36} T^{21} - 10068774 p^{40} T^{22} - 24 p^{44} T^{23} + p^{48} T^{24} \)
53 \( 1 - 414 T - 13725134 T^{2} - 7014732824 T^{3} + 165958055930181 T^{4} + 208417442032136246 T^{5} - \)\(14\!\cdots\!44\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!16\)\( T^{8} + \)\(14\!\cdots\!66\)\( T^{9} - \)\(14\!\cdots\!58\)\( p T^{10} - \)\(33\!\cdots\!74\)\( T^{11} + \)\(54\!\cdots\!16\)\( T^{12} - \)\(33\!\cdots\!74\)\( p^{4} T^{13} - \)\(14\!\cdots\!58\)\( p^{9} T^{14} + \)\(14\!\cdots\!66\)\( p^{12} T^{15} + \)\(12\!\cdots\!16\)\( p^{16} T^{16} - \)\(22\!\cdots\!24\)\( p^{20} T^{17} - \)\(14\!\cdots\!44\)\( p^{24} T^{18} + 208417442032136246 p^{28} T^{19} + 165958055930181 p^{32} T^{20} - 7014732824 p^{36} T^{21} - 13725134 p^{40} T^{22} - 414 p^{44} T^{23} + p^{48} T^{24} \)
59 \( 1 + 10011 T + 24606746 T^{2} - 48335046044 T^{3} - 85055058906234 T^{4} + 1848432652355091201 T^{5} + \)\(68\!\cdots\!21\)\( T^{6} - \)\(10\!\cdots\!34\)\( T^{7} - \)\(82\!\cdots\!39\)\( T^{8} + \)\(15\!\cdots\!31\)\( T^{9} + \)\(13\!\cdots\!41\)\( T^{10} - \)\(16\!\cdots\!29\)\( T^{11} - \)\(23\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!29\)\( p^{4} T^{13} + \)\(13\!\cdots\!41\)\( p^{8} T^{14} + \)\(15\!\cdots\!31\)\( p^{12} T^{15} - \)\(82\!\cdots\!39\)\( p^{16} T^{16} - \)\(10\!\cdots\!34\)\( p^{20} T^{17} + \)\(68\!\cdots\!21\)\( p^{24} T^{18} + 1848432652355091201 p^{28} T^{19} - 85055058906234 p^{32} T^{20} - 48335046044 p^{36} T^{21} + 24606746 p^{40} T^{22} + 10011 p^{44} T^{23} + p^{48} T^{24} \)
61 \( 1 - 9460 T + 57744648 T^{2} - 309062215210 T^{3} + 1619001804085751 T^{4} - 7693654169306140700 T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!60\)\( T^{8} - \)\(32\!\cdots\!20\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} - \)\(48\!\cdots\!10\)\( T^{11} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(48\!\cdots\!10\)\( p^{4} T^{13} + \)\(12\!\cdots\!28\)\( p^{8} T^{14} - \)\(32\!\cdots\!20\)\( p^{12} T^{15} + \)\(77\!\cdots\!60\)\( p^{16} T^{16} - \)\(16\!\cdots\!00\)\( p^{20} T^{17} + \)\(35\!\cdots\!40\)\( p^{24} T^{18} - 7693654169306140700 p^{28} T^{19} + 1619001804085751 p^{32} T^{20} - 309062215210 p^{36} T^{21} + 57744648 p^{40} T^{22} - 9460 p^{44} T^{23} + p^{48} T^{24} \)
67 \( ( 1 - 6077 T + 35718991 T^{2} - 18219600105 T^{3} + 122534918608155 T^{4} + 2701855134548306278 T^{5} - \)\(61\!\cdots\!46\)\( T^{6} + 2701855134548306278 p^{4} T^{7} + 122534918608155 p^{8} T^{8} - 18219600105 p^{12} T^{9} + 35718991 p^{16} T^{10} - 6077 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
71 \( 1 - 17574 T + 104286776 T^{2} - 424728504824 T^{3} + 5029513991381961 T^{4} - 44618257451712179514 T^{5} + \)\(20\!\cdots\!06\)\( T^{6} - \)\(10\!\cdots\!44\)\( T^{7} + \)\(80\!\cdots\!46\)\( T^{8} - \)\(47\!\cdots\!54\)\( T^{9} + \)\(20\!\cdots\!26\)\( T^{10} - \)\(11\!\cdots\!94\)\( T^{11} + \)\(68\!\cdots\!36\)\( T^{12} - \)\(11\!\cdots\!94\)\( p^{4} T^{13} + \)\(20\!\cdots\!26\)\( p^{8} T^{14} - \)\(47\!\cdots\!54\)\( p^{12} T^{15} + \)\(80\!\cdots\!46\)\( p^{16} T^{16} - \)\(10\!\cdots\!44\)\( p^{20} T^{17} + \)\(20\!\cdots\!06\)\( p^{24} T^{18} - 44618257451712179514 p^{28} T^{19} + 5029513991381961 p^{32} T^{20} - 424728504824 p^{36} T^{21} + 104286776 p^{40} T^{22} - 17574 p^{44} T^{23} + p^{48} T^{24} \)
73 \( 1 - 27950 T + 454822888 T^{2} - 5337429236470 T^{3} + 50136942858006191 T^{4} - \)\(39\!\cdots\!10\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} - \)\(17\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!60\)\( T^{8} - \)\(54\!\cdots\!10\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} - \)\(15\!\cdots\!80\)\( T^{11} + \)\(84\!\cdots\!04\)\( T^{12} - \)\(15\!\cdots\!80\)\( p^{4} T^{13} + \)\(29\!\cdots\!08\)\( p^{8} T^{14} - \)\(54\!\cdots\!10\)\( p^{12} T^{15} + \)\(99\!\cdots\!60\)\( p^{16} T^{16} - \)\(17\!\cdots\!80\)\( p^{20} T^{17} + \)\(27\!\cdots\!20\)\( p^{24} T^{18} - \)\(39\!\cdots\!10\)\( p^{28} T^{19} + 50136942858006191 p^{32} T^{20} - 5337429236470 p^{36} T^{21} + 454822888 p^{40} T^{22} - 27950 p^{44} T^{23} + p^{48} T^{24} \)
79 \( 1 + 41540 T + 884578158 T^{2} + 13528269571760 T^{3} + 171031570391119991 T^{4} + \)\(18\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!10\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(82\!\cdots\!08\)\( T^{10} + \)\(56\!\cdots\!20\)\( T^{11} + \)\(36\!\cdots\!24\)\( T^{12} + \)\(56\!\cdots\!20\)\( p^{4} T^{13} + \)\(82\!\cdots\!08\)\( p^{8} T^{14} + \)\(11\!\cdots\!40\)\( p^{12} T^{15} + \)\(14\!\cdots\!70\)\( p^{16} T^{16} + \)\(17\!\cdots\!80\)\( p^{20} T^{17} + \)\(18\!\cdots\!10\)\( p^{24} T^{18} + \)\(18\!\cdots\!60\)\( p^{28} T^{19} + 171031570391119991 p^{32} T^{20} + 13528269571760 p^{36} T^{21} + 884578158 p^{40} T^{22} + 41540 p^{44} T^{23} + p^{48} T^{24} \)
83 \( 1 + 18665 T + 224643628 T^{2} + 792700664890 T^{3} - 7907731388500564 T^{4} - \)\(17\!\cdots\!75\)\( T^{5} - \)\(12\!\cdots\!65\)\( T^{6} - \)\(28\!\cdots\!70\)\( T^{7} + \)\(49\!\cdots\!05\)\( T^{8} + \)\(50\!\cdots\!25\)\( T^{9} + \)\(19\!\cdots\!53\)\( T^{10} - \)\(10\!\cdots\!35\)\( T^{11} - \)\(13\!\cdots\!56\)\( T^{12} - \)\(10\!\cdots\!35\)\( p^{4} T^{13} + \)\(19\!\cdots\!53\)\( p^{8} T^{14} + \)\(50\!\cdots\!25\)\( p^{12} T^{15} + \)\(49\!\cdots\!05\)\( p^{16} T^{16} - \)\(28\!\cdots\!70\)\( p^{20} T^{17} - \)\(12\!\cdots\!65\)\( p^{24} T^{18} - \)\(17\!\cdots\!75\)\( p^{28} T^{19} - 7907731388500564 p^{32} T^{20} + 792700664890 p^{36} T^{21} + 224643628 p^{40} T^{22} + 18665 p^{44} T^{23} + p^{48} T^{24} \)
89 \( ( 1 - 2777 T + 289401541 T^{2} - 984806913445 T^{3} + 37686355645869335 T^{4} - \)\(12\!\cdots\!62\)\( T^{5} + \)\(29\!\cdots\!14\)\( T^{6} - \)\(12\!\cdots\!62\)\( p^{4} T^{7} + 37686355645869335 p^{8} T^{8} - 984806913445 p^{12} T^{9} + 289401541 p^{16} T^{10} - 2777 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
97 \( 1 - 20769 T + 424679856 T^{2} - 8132949547004 T^{3} + 119819968442578026 T^{4} - \)\(16\!\cdots\!09\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} - \)\(25\!\cdots\!44\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} - \)\(31\!\cdots\!79\)\( T^{9} + \)\(31\!\cdots\!81\)\( T^{10} - \)\(31\!\cdots\!59\)\( T^{11} + \)\(30\!\cdots\!76\)\( T^{12} - \)\(31\!\cdots\!59\)\( p^{4} T^{13} + \)\(31\!\cdots\!81\)\( p^{8} T^{14} - \)\(31\!\cdots\!79\)\( p^{12} T^{15} + \)\(28\!\cdots\!01\)\( p^{16} T^{16} - \)\(25\!\cdots\!44\)\( p^{20} T^{17} + \)\(21\!\cdots\!91\)\( p^{24} T^{18} - \)\(16\!\cdots\!09\)\( p^{28} T^{19} + 119819968442578026 p^{32} T^{20} - 8132949547004 p^{36} T^{21} + 424679856 p^{40} T^{22} - 20769 p^{44} T^{23} + p^{48} T^{24} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.984619358551889532129356863252, −7.957421545799656679234928090308, −7.55148709967729826471510068919, −7.42966270034087887983702054793, −7.15680928599117016741818066737, −6.94001776307360107486864055567, −6.79519065506185683550935256931, −6.71362726836937285063502453491, −6.70588737493782200969618209835, −6.28960553433772509791866433840, −6.24415329828003939825929847300, −6.18891829672656497768477447539, −5.33375273399099302973235601308, −5.16504122985357109131641935826, −5.13897436698524551184570730890, −5.12814304516471770097483648457, −4.41089944555135701446353188788, −4.34097181921691043144127527094, −3.99843181538594636360595410056, −3.78568739047174893620691187189, −3.68376350884209958541191495627, −3.06720870789742373251382605435, −2.30452235969826111214534811765, −1.78430449216247841846235652431, −0.04768232989740300483805046856, 0.04768232989740300483805046856, 1.78430449216247841846235652431, 2.30452235969826111214534811765, 3.06720870789742373251382605435, 3.68376350884209958541191495627, 3.78568739047174893620691187189, 3.99843181538594636360595410056, 4.34097181921691043144127527094, 4.41089944555135701446353188788, 5.12814304516471770097483648457, 5.13897436698524551184570730890, 5.16504122985357109131641935826, 5.33375273399099302973235601308, 6.18891829672656497768477447539, 6.24415329828003939825929847300, 6.28960553433772509791866433840, 6.70588737493782200969618209835, 6.71362726836937285063502453491, 6.79519065506185683550935256931, 6.94001776307360107486864055567, 7.15680928599117016741818066737, 7.42966270034087887983702054793, 7.55148709967729826471510068919, 7.957421545799656679234928090308, 7.984619358551889532129356863252

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

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