Properties

Degree 26
Conductor $ 43^{13} $
Sign $1$
Motivic weight 7
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 94·3-s − 243·4-s + 998·5-s + 1.50e3·6-s + 1.36e3·7-s − 4.64e3·8-s − 4.79e3·9-s + 1.59e4·10-s + 1.62e3·11-s − 2.28e4·12-s + 1.35e4·13-s + 2.17e4·14-s + 9.38e4·15-s + 3.27e4·16-s + 1.10e5·17-s − 7.66e4·18-s + 1.05e5·19-s − 2.42e5·20-s + 1.27e5·21-s + 2.59e4·22-s + 1.60e5·23-s − 4.36e5·24-s + 1.25e5·25-s + 2.16e5·26-s − 7.11e5·27-s − 3.30e5·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.01·3-s − 1.89·4-s + 3.57·5-s + 2.84·6-s + 1.49·7-s − 3.20·8-s − 2.19·9-s + 5.04·10-s + 0.366·11-s − 3.81·12-s + 1.71·13-s + 2.11·14-s + 7.17·15-s + 2.00·16-s + 5.47·17-s − 3.09·18-s + 3.51·19-s − 6.77·20-s + 3.01·21-s + 0.518·22-s + 2.74·23-s − 6.44·24-s + 1.60·25-s + 2.41·26-s − 6.95·27-s − 2.84·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{13}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(26\)
\( N \)  =  \(43^{13}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((26,\ 43^{13} ,\ ( \ : [7/2]^{13} ),\ 1 )\)
\(L(4)\)  \(\approx\)  \(35.8257\)
\(L(\frac12)\)  \(\approx\)  \(35.8257\)
\(L(\frac{9}{2})\)   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 26. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 25.
$p$$F_p(T)$
bad43 \( ( 1 + p^{3} T )^{13} \)
good2 \( 1 - p^{4} T + 499 T^{2} - 3613 p T^{3} + 64877 p T^{4} - 405777 p^{2} T^{5} + 6265495 p^{2} T^{6} - 37718987 p^{3} T^{7} + 135023115 p^{5} T^{8} - 837154915 p^{6} T^{9} + 10730204725 p^{6} T^{10} - 64273522057 p^{7} T^{11} + 23846127347 p^{12} T^{12} - 268855436977 p^{12} T^{13} + 23846127347 p^{19} T^{14} - 64273522057 p^{21} T^{15} + 10730204725 p^{27} T^{16} - 837154915 p^{34} T^{17} + 135023115 p^{40} T^{18} - 37718987 p^{45} T^{19} + 6265495 p^{51} T^{20} - 405777 p^{58} T^{21} + 64877 p^{64} T^{22} - 3613 p^{71} T^{23} + 499 p^{77} T^{24} - p^{88} T^{25} + p^{91} T^{26} \)
3 \( 1 - 94 T + 13628 T^{2} - 113314 p^{2} T^{3} + 84521770 T^{4} - 4911562460 T^{5} + 301148239904 T^{6} - 4518316633510 p T^{7} + 8234084195492 p^{4} T^{8} - 827623451819782 p^{3} T^{9} + 11650378306251005 p^{4} T^{10} - 27711179995602220 p^{6} T^{11} + 1327869257067469910 p^{6} T^{12} - 6829120720967041816 p^{7} T^{13} + 1327869257067469910 p^{13} T^{14} - 27711179995602220 p^{20} T^{15} + 11650378306251005 p^{25} T^{16} - 827623451819782 p^{31} T^{17} + 8234084195492 p^{39} T^{18} - 4518316633510 p^{43} T^{19} + 301148239904 p^{49} T^{20} - 4911562460 p^{56} T^{21} + 84521770 p^{63} T^{22} - 113314 p^{72} T^{23} + 13628 p^{77} T^{24} - 94 p^{84} T^{25} + p^{91} T^{26} \)
5 \( 1 - 998 T + 174148 p T^{2} - 535689006 T^{3} + 300509982542 T^{4} - 28693425481694 p T^{5} + 2560442089648656 p^{2} T^{6} - 206419755308243666 p^{3} T^{7} + 15755669023851882384 p^{4} T^{8} - \)\(11\!\cdots\!22\)\( p^{5} T^{9} + \)\(75\!\cdots\!99\)\( p^{6} T^{10} - \)\(47\!\cdots\!32\)\( p^{7} T^{11} + \)\(28\!\cdots\!26\)\( p^{8} T^{12} - \)\(16\!\cdots\!08\)\( p^{9} T^{13} + \)\(28\!\cdots\!26\)\( p^{15} T^{14} - \)\(47\!\cdots\!32\)\( p^{21} T^{15} + \)\(75\!\cdots\!99\)\( p^{27} T^{16} - \)\(11\!\cdots\!22\)\( p^{33} T^{17} + 15755669023851882384 p^{39} T^{18} - 206419755308243666 p^{45} T^{19} + 2560442089648656 p^{51} T^{20} - 28693425481694 p^{57} T^{21} + 300509982542 p^{63} T^{22} - 535689006 p^{70} T^{23} + 174148 p^{78} T^{24} - 998 p^{84} T^{25} + p^{91} T^{26} \)
7 \( 1 - 1360 T + 7058509 T^{2} - 7851071624 T^{3} + 22977827763868 T^{4} - 21067270496738864 T^{5} + 6663944414069580156 p T^{6} - \)\(35\!\cdots\!92\)\( T^{7} + \)\(68\!\cdots\!45\)\( T^{8} - \)\(44\!\cdots\!60\)\( T^{9} + \)\(78\!\cdots\!53\)\( T^{10} - \)\(44\!\cdots\!96\)\( T^{11} + \)\(74\!\cdots\!48\)\( T^{12} - \)\(56\!\cdots\!76\)\( p T^{13} + \)\(74\!\cdots\!48\)\( p^{7} T^{14} - \)\(44\!\cdots\!96\)\( p^{14} T^{15} + \)\(78\!\cdots\!53\)\( p^{21} T^{16} - \)\(44\!\cdots\!60\)\( p^{28} T^{17} + \)\(68\!\cdots\!45\)\( p^{35} T^{18} - \)\(35\!\cdots\!92\)\( p^{42} T^{19} + 6663944414069580156 p^{50} T^{20} - 21067270496738864 p^{56} T^{21} + 22977827763868 p^{63} T^{22} - 7851071624 p^{70} T^{23} + 7058509 p^{77} T^{24} - 1360 p^{84} T^{25} + p^{91} T^{26} \)
11 \( 1 - 1620 T + 132588019 T^{2} - 123981604500 T^{3} + 7703760283965880 T^{4} - 3713200717818796164 T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!44\)\( T^{8} - \)\(83\!\cdots\!32\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} - \)\(37\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} - \)\(87\!\cdots\!20\)\( p T^{13} + \)\(11\!\cdots\!86\)\( p^{7} T^{14} - \)\(37\!\cdots\!84\)\( p^{14} T^{15} + \)\(85\!\cdots\!00\)\( p^{21} T^{16} - \)\(83\!\cdots\!32\)\( p^{28} T^{17} + \)\(55\!\cdots\!44\)\( p^{35} T^{18} - \)\(12\!\cdots\!20\)\( p^{42} T^{19} + \)\(25\!\cdots\!00\)\( p^{49} T^{20} - 3713200717818796164 p^{56} T^{21} + 7703760283965880 p^{63} T^{22} - 123981604500 p^{70} T^{23} + 132588019 p^{77} T^{24} - 1620 p^{84} T^{25} + p^{91} T^{26} \)
13 \( 1 - 13550 T + 451004117 T^{2} - 4457271716692 T^{3} + 93112309019999992 T^{4} - \)\(74\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} - \)\(87\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!96\)\( T^{8} - \)\(81\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!86\)\( T^{12} - \)\(42\!\cdots\!16\)\( T^{13} + \)\(82\!\cdots\!86\)\( p^{7} T^{14} - \)\(63\!\cdots\!80\)\( p^{14} T^{15} + \)\(11\!\cdots\!04\)\( p^{21} T^{16} - \)\(81\!\cdots\!96\)\( p^{28} T^{17} + \)\(13\!\cdots\!96\)\( p^{35} T^{18} - \)\(87\!\cdots\!48\)\( p^{42} T^{19} + \)\(12\!\cdots\!04\)\( p^{49} T^{20} - \)\(74\!\cdots\!76\)\( p^{56} T^{21} + 93112309019999992 p^{63} T^{22} - 4457271716692 p^{70} T^{23} + 451004117 p^{77} T^{24} - 13550 p^{84} T^{25} + p^{91} T^{26} \)
17 \( 1 - 110880 T + 7269241380 T^{2} - 350308303802712 T^{3} + 13915691873194028506 T^{4} - \)\(48\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!91\)\( T^{8} - \)\(30\!\cdots\!04\)\( T^{9} + \)\(72\!\cdots\!29\)\( T^{10} - \)\(16\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} - \)\(74\!\cdots\!34\)\( T^{13} + \)\(35\!\cdots\!06\)\( p^{7} T^{14} - \)\(16\!\cdots\!04\)\( p^{14} T^{15} + \)\(72\!\cdots\!29\)\( p^{21} T^{16} - \)\(30\!\cdots\!04\)\( p^{28} T^{17} + \)\(11\!\cdots\!91\)\( p^{35} T^{18} - \)\(43\!\cdots\!00\)\( p^{42} T^{19} + \)\(15\!\cdots\!04\)\( p^{49} T^{20} - \)\(48\!\cdots\!14\)\( p^{56} T^{21} + 13915691873194028506 p^{63} T^{22} - 350308303802712 p^{70} T^{23} + 7269241380 p^{77} T^{24} - 110880 p^{84} T^{25} + p^{91} T^{26} \)
19 \( 1 - 105058 T + 10894929034 T^{2} - 729074101697894 T^{3} + 46770979546573980358 T^{4} - \)\(24\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!42\)\( T^{6} - \)\(27\!\cdots\!94\)\( p T^{7} + \)\(22\!\cdots\!20\)\( T^{8} - \)\(85\!\cdots\!82\)\( T^{9} + \)\(31\!\cdots\!91\)\( T^{10} - \)\(10\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!64\)\( p^{7} T^{14} - \)\(10\!\cdots\!04\)\( p^{14} T^{15} + \)\(31\!\cdots\!91\)\( p^{21} T^{16} - \)\(85\!\cdots\!82\)\( p^{28} T^{17} + \)\(22\!\cdots\!20\)\( p^{35} T^{18} - \)\(27\!\cdots\!94\)\( p^{43} T^{19} + \)\(12\!\cdots\!42\)\( p^{49} T^{20} - \)\(24\!\cdots\!36\)\( p^{56} T^{21} + 46770979546573980358 p^{63} T^{22} - 729074101697894 p^{70} T^{23} + 10894929034 p^{77} T^{24} - 105058 p^{84} T^{25} + p^{91} T^{26} \)
23 \( 1 - 160184 T + 36979701828 T^{2} - 4753092322433576 T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(68\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!92\)\( T^{6} - \)\(64\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!51\)\( T^{8} - \)\(42\!\cdots\!96\)\( T^{9} + \)\(13\!\cdots\!05\)\( p T^{10} - \)\(21\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} - \)\(83\!\cdots\!40\)\( T^{13} + \)\(13\!\cdots\!02\)\( p^{7} T^{14} - \)\(21\!\cdots\!56\)\( p^{14} T^{15} + \)\(13\!\cdots\!05\)\( p^{22} T^{16} - \)\(42\!\cdots\!96\)\( p^{28} T^{17} + \)\(55\!\cdots\!51\)\( p^{35} T^{18} - \)\(64\!\cdots\!20\)\( p^{42} T^{19} + \)\(71\!\cdots\!92\)\( p^{49} T^{20} - \)\(68\!\cdots\!64\)\( p^{56} T^{21} + \)\(65\!\cdots\!70\)\( p^{63} T^{22} - 4753092322433576 p^{70} T^{23} + 36979701828 p^{77} T^{24} - 160184 p^{84} T^{25} + p^{91} T^{26} \)
29 \( 1 - 285546 T + 180604853400 T^{2} - 41013344500675266 T^{3} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(27\!\cdots\!34\)\( T^{5} + \)\(71\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!24\)\( T^{8} - \)\(32\!\cdots\!66\)\( T^{9} + \)\(61\!\cdots\!47\)\( T^{10} - \)\(73\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!62\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!62\)\( p^{7} T^{14} - \)\(73\!\cdots\!48\)\( p^{14} T^{15} + \)\(61\!\cdots\!47\)\( p^{21} T^{16} - \)\(32\!\cdots\!66\)\( p^{28} T^{17} + \)\(24\!\cdots\!24\)\( p^{35} T^{18} - \)\(11\!\cdots\!42\)\( p^{42} T^{19} + \)\(71\!\cdots\!40\)\( p^{49} T^{20} - \)\(27\!\cdots\!34\)\( p^{56} T^{21} + \)\(14\!\cdots\!66\)\( p^{63} T^{22} - 41013344500675266 p^{70} T^{23} + 180604853400 p^{77} T^{24} - 285546 p^{84} T^{25} + p^{91} T^{26} \)
31 \( 1 + 99616 T + 241614413464 T^{2} + 18338344889599376 T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!92\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!72\)\( T^{9} + \)\(42\!\cdots\!31\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(14\!\cdots\!42\)\( p^{7} T^{14} + \)\(15\!\cdots\!08\)\( p^{14} T^{15} + \)\(42\!\cdots\!31\)\( p^{21} T^{16} + \)\(16\!\cdots\!72\)\( p^{28} T^{17} + \)\(10\!\cdots\!63\)\( p^{35} T^{18} + \)\(65\!\cdots\!12\)\( p^{42} T^{19} + \)\(20\!\cdots\!92\)\( p^{49} T^{20} + \)\(14\!\cdots\!12\)\( p^{56} T^{21} + \)\(27\!\cdots\!10\)\( p^{63} T^{22} + 18338344889599376 p^{70} T^{23} + 241614413464 p^{77} T^{24} + 99616 p^{84} T^{25} + p^{91} T^{26} \)
37 \( 1 - 176038 T + 1002592896582 T^{2} - 172621415570488610 T^{3} + \)\(48\!\cdots\!62\)\( T^{4} - \)\(79\!\cdots\!90\)\( T^{5} + \)\(14\!\cdots\!62\)\( T^{6} - \)\(22\!\cdots\!06\)\( T^{7} + \)\(31\!\cdots\!64\)\( T^{8} - \)\(45\!\cdots\!18\)\( T^{9} + \)\(50\!\cdots\!45\)\( T^{10} - \)\(65\!\cdots\!96\)\( T^{11} + \)\(61\!\cdots\!44\)\( T^{12} - \)\(71\!\cdots\!12\)\( T^{13} + \)\(61\!\cdots\!44\)\( p^{7} T^{14} - \)\(65\!\cdots\!96\)\( p^{14} T^{15} + \)\(50\!\cdots\!45\)\( p^{21} T^{16} - \)\(45\!\cdots\!18\)\( p^{28} T^{17} + \)\(31\!\cdots\!64\)\( p^{35} T^{18} - \)\(22\!\cdots\!06\)\( p^{42} T^{19} + \)\(14\!\cdots\!62\)\( p^{49} T^{20} - \)\(79\!\cdots\!90\)\( p^{56} T^{21} + \)\(48\!\cdots\!62\)\( p^{63} T^{22} - 172621415570488610 p^{70} T^{23} + 1002592896582 p^{77} T^{24} - 176038 p^{84} T^{25} + p^{91} T^{26} \)
41 \( 1 + 410260 T + 31116447684 p T^{2} + 520566650539856152 T^{3} + \)\(84\!\cdots\!34\)\( T^{4} + \)\(33\!\cdots\!10\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!59\)\( T^{8} + \)\(44\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!41\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{11} + \)\(73\!\cdots\!58\)\( T^{12} + \)\(23\!\cdots\!38\)\( T^{13} + \)\(73\!\cdots\!58\)\( p^{7} T^{14} + \)\(11\!\cdots\!44\)\( p^{14} T^{15} + \)\(32\!\cdots\!41\)\( p^{21} T^{16} + \)\(44\!\cdots\!08\)\( p^{28} T^{17} + \)\(12\!\cdots\!59\)\( p^{35} T^{18} + \)\(14\!\cdots\!96\)\( p^{42} T^{19} + \)\(37\!\cdots\!04\)\( p^{49} T^{20} + \)\(33\!\cdots\!10\)\( p^{56} T^{21} + \)\(84\!\cdots\!34\)\( p^{63} T^{22} + 520566650539856152 p^{70} T^{23} + 31116447684 p^{78} T^{24} + 410260 p^{84} T^{25} + p^{91} T^{26} \)
47 \( 1 + 424556 T + 3688252683440 T^{2} + 2420585272729575804 T^{3} + \)\(67\!\cdots\!76\)\( T^{4} + \)\(56\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!12\)\( T^{6} + \)\(77\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!18\)\( T^{8} + \)\(74\!\cdots\!56\)\( T^{9} + \)\(64\!\cdots\!19\)\( T^{10} + \)\(53\!\cdots\!64\)\( T^{11} + \)\(40\!\cdots\!52\)\( T^{12} + \)\(30\!\cdots\!32\)\( T^{13} + \)\(40\!\cdots\!52\)\( p^{7} T^{14} + \)\(53\!\cdots\!64\)\( p^{14} T^{15} + \)\(64\!\cdots\!19\)\( p^{21} T^{16} + \)\(74\!\cdots\!56\)\( p^{28} T^{17} + \)\(81\!\cdots\!18\)\( p^{35} T^{18} + \)\(77\!\cdots\!80\)\( p^{42} T^{19} + \)\(83\!\cdots\!12\)\( p^{49} T^{20} + \)\(56\!\cdots\!60\)\( p^{56} T^{21} + \)\(67\!\cdots\!76\)\( p^{63} T^{22} + 2420585272729575804 p^{70} T^{23} + 3688252683440 p^{77} T^{24} + 424556 p^{84} T^{25} + p^{91} T^{26} \)
53 \( 1 - 3992458 T + 15589250135817 T^{2} - 39082970394967823704 T^{3} + \)\(92\!\cdots\!16\)\( T^{4} - \)\(17\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} - \)\(49\!\cdots\!88\)\( T^{7} + \)\(73\!\cdots\!96\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} - \)\(16\!\cdots\!28\)\( T^{11} + \)\(19\!\cdots\!82\)\( T^{12} - \)\(21\!\cdots\!60\)\( T^{13} + \)\(19\!\cdots\!82\)\( p^{7} T^{14} - \)\(16\!\cdots\!28\)\( p^{14} T^{15} + \)\(13\!\cdots\!44\)\( p^{21} T^{16} - \)\(10\!\cdots\!68\)\( p^{28} T^{17} + \)\(73\!\cdots\!96\)\( p^{35} T^{18} - \)\(49\!\cdots\!88\)\( p^{42} T^{19} + \)\(31\!\cdots\!44\)\( p^{49} T^{20} - \)\(17\!\cdots\!84\)\( p^{56} T^{21} + \)\(92\!\cdots\!16\)\( p^{63} T^{22} - 39082970394967823704 p^{70} T^{23} + 15589250135817 p^{77} T^{24} - 3992458 p^{84} T^{25} + p^{91} T^{26} \)
59 \( 1 - 2248836 T + 18986189218171 T^{2} - 636896445896421216 p T^{3} + \)\(17\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!54\)\( T^{6} - \)\(17\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!59\)\( T^{8} - \)\(76\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!17\)\( T^{10} - \)\(25\!\cdots\!88\)\( T^{11} + \)\(56\!\cdots\!40\)\( T^{12} - \)\(70\!\cdots\!64\)\( T^{13} + \)\(56\!\cdots\!40\)\( p^{7} T^{14} - \)\(25\!\cdots\!88\)\( p^{14} T^{15} + \)\(18\!\cdots\!17\)\( p^{21} T^{16} - \)\(76\!\cdots\!40\)\( p^{28} T^{17} + \)\(51\!\cdots\!59\)\( p^{35} T^{18} - \)\(17\!\cdots\!32\)\( p^{42} T^{19} + \)\(11\!\cdots\!54\)\( p^{49} T^{20} - \)\(31\!\cdots\!68\)\( p^{56} T^{21} + \)\(17\!\cdots\!18\)\( p^{63} T^{22} - 636896445896421216 p^{71} T^{23} + 18986189218171 p^{77} T^{24} - 2248836 p^{84} T^{25} + p^{91} T^{26} \)
61 \( 1 - 6210394 T + 38426666503003 T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!92\)\( T^{5} + \)\(56\!\cdots\!64\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!17\)\( T^{8} - \)\(83\!\cdots\!78\)\( T^{9} + \)\(18\!\cdots\!55\)\( T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{13} + \)\(70\!\cdots\!72\)\( p^{7} T^{14} - \)\(36\!\cdots\!04\)\( p^{14} T^{15} + \)\(18\!\cdots\!55\)\( p^{21} T^{16} - \)\(83\!\cdots\!78\)\( p^{28} T^{17} + \)\(36\!\cdots\!17\)\( p^{35} T^{18} - \)\(14\!\cdots\!80\)\( p^{42} T^{19} + \)\(56\!\cdots\!64\)\( p^{49} T^{20} - \)\(18\!\cdots\!92\)\( p^{56} T^{21} + \)\(60\!\cdots\!88\)\( p^{63} T^{22} - \)\(15\!\cdots\!76\)\( p^{70} T^{23} + 38426666503003 p^{77} T^{24} - 6210394 p^{84} T^{25} + p^{91} T^{26} \)
67 \( 1 + 1993648 T + 36332499483043 T^{2} + 80369020063411411764 T^{3} + \)\(68\!\cdots\!56\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(90\!\cdots\!36\)\( T^{8} + \)\(22\!\cdots\!72\)\( T^{9} + \)\(75\!\cdots\!16\)\( T^{10} + \)\(18\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!18\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{13} + \)\(53\!\cdots\!18\)\( p^{7} T^{14} + \)\(18\!\cdots\!92\)\( p^{14} T^{15} + \)\(75\!\cdots\!16\)\( p^{21} T^{16} + \)\(22\!\cdots\!72\)\( p^{28} T^{17} + \)\(90\!\cdots\!36\)\( p^{35} T^{18} + \)\(22\!\cdots\!60\)\( p^{42} T^{19} + \)\(89\!\cdots\!48\)\( p^{49} T^{20} + \)\(16\!\cdots\!32\)\( p^{56} T^{21} + \)\(68\!\cdots\!56\)\( p^{63} T^{22} + 80369020063411411764 p^{70} T^{23} + 36332499483043 p^{77} T^{24} + 1993648 p^{84} T^{25} + p^{91} T^{26} \)
71 \( 1 - 4978064 T + 87202612193659 T^{2} - 5528935375628066240 p T^{3} + \)\(37\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!70\)\( T^{6} - \)\(37\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!19\)\( T^{8} - \)\(66\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!13\)\( T^{10} - \)\(88\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!08\)\( T^{12} - \)\(91\!\cdots\!08\)\( T^{13} + \)\(34\!\cdots\!08\)\( p^{7} T^{14} - \)\(88\!\cdots\!48\)\( p^{14} T^{15} + \)\(29\!\cdots\!13\)\( p^{21} T^{16} - \)\(66\!\cdots\!92\)\( p^{28} T^{17} + \)\(20\!\cdots\!19\)\( p^{35} T^{18} - \)\(37\!\cdots\!88\)\( p^{42} T^{19} + \)\(10\!\cdots\!70\)\( p^{49} T^{20} - \)\(15\!\cdots\!52\)\( p^{56} T^{21} + \)\(37\!\cdots\!18\)\( p^{63} T^{22} - 5528935375628066240 p^{71} T^{23} + 87202612193659 p^{77} T^{24} - 4978064 p^{84} T^{25} + p^{91} T^{26} \)
73 \( 1 - 8224814 T + 92481350221839 T^{2} - \)\(53\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!04\)\( T^{4} - \)\(18\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} - \)\(85\!\cdots\!50\)\( T^{9} + \)\(36\!\cdots\!23\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} + \)\(49\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + \)\(49\!\cdots\!68\)\( p^{7} T^{14} - \)\(12\!\cdots\!68\)\( p^{14} T^{15} + \)\(36\!\cdots\!23\)\( p^{21} T^{16} - \)\(85\!\cdots\!50\)\( p^{28} T^{17} + \)\(22\!\cdots\!37\)\( p^{35} T^{18} - \)\(45\!\cdots\!64\)\( p^{42} T^{19} + \)\(10\!\cdots\!88\)\( p^{49} T^{20} - \)\(18\!\cdots\!96\)\( p^{56} T^{21} + \)\(38\!\cdots\!04\)\( p^{63} T^{22} - \)\(53\!\cdots\!16\)\( p^{70} T^{23} + 92481350221839 p^{77} T^{24} - 8224814 p^{84} T^{25} + p^{91} T^{26} \)
79 \( 1 - 6945708 T + 117121136837052 T^{2} - \)\(56\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(72\!\cdots\!74\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + \)\(18\!\cdots\!35\)\( T^{10} - \)\(46\!\cdots\!04\)\( T^{11} + \)\(42\!\cdots\!92\)\( T^{12} - \)\(97\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!92\)\( p^{7} T^{14} - \)\(46\!\cdots\!04\)\( p^{14} T^{15} + \)\(18\!\cdots\!35\)\( p^{21} T^{16} - \)\(19\!\cdots\!48\)\( p^{28} T^{17} + \)\(72\!\cdots\!74\)\( p^{35} T^{18} - \)\(74\!\cdots\!48\)\( p^{42} T^{19} + \)\(24\!\cdots\!68\)\( p^{49} T^{20} - \)\(24\!\cdots\!60\)\( p^{56} T^{21} + \)\(65\!\cdots\!68\)\( p^{63} T^{22} - \)\(56\!\cdots\!28\)\( p^{70} T^{23} + 117121136837052 p^{77} T^{24} - 6945708 p^{84} T^{25} + p^{91} T^{26} \)
83 \( 1 - 22937328 T + 403848212345231 T^{2} - \)\(46\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} - \)\(35\!\cdots\!72\)\( T^{5} + \)\(24\!\cdots\!76\)\( T^{6} - \)\(14\!\cdots\!68\)\( T^{7} + \)\(83\!\cdots\!60\)\( T^{8} - \)\(44\!\cdots\!24\)\( T^{9} + \)\(27\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!32\)\( T^{11} + \)\(97\!\cdots\!38\)\( T^{12} - \)\(51\!\cdots\!32\)\( T^{13} + \)\(97\!\cdots\!38\)\( p^{7} T^{14} - \)\(15\!\cdots\!32\)\( p^{14} T^{15} + \)\(27\!\cdots\!48\)\( p^{21} T^{16} - \)\(44\!\cdots\!24\)\( p^{28} T^{17} + \)\(83\!\cdots\!60\)\( p^{35} T^{18} - \)\(14\!\cdots\!68\)\( p^{42} T^{19} + \)\(24\!\cdots\!76\)\( p^{49} T^{20} - \)\(35\!\cdots\!72\)\( p^{56} T^{21} + \)\(46\!\cdots\!36\)\( p^{63} T^{22} - \)\(46\!\cdots\!40\)\( p^{70} T^{23} + 403848212345231 p^{77} T^{24} - 22937328 p^{84} T^{25} + p^{91} T^{26} \)
89 \( 1 - 9291302 T + 323406546094099 T^{2} - \)\(31\!\cdots\!04\)\( T^{3} + \)\(56\!\cdots\!92\)\( T^{4} - \)\(51\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!68\)\( T^{6} - \)\(56\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!61\)\( T^{8} - \)\(45\!\cdots\!46\)\( T^{9} + \)\(42\!\cdots\!63\)\( T^{10} - \)\(28\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!60\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!60\)\( p^{7} T^{14} - \)\(28\!\cdots\!48\)\( p^{14} T^{15} + \)\(42\!\cdots\!63\)\( p^{21} T^{16} - \)\(45\!\cdots\!46\)\( p^{28} T^{17} + \)\(61\!\cdots\!61\)\( p^{35} T^{18} - \)\(56\!\cdots\!24\)\( p^{42} T^{19} + \)\(68\!\cdots\!68\)\( p^{49} T^{20} - \)\(51\!\cdots\!56\)\( p^{56} T^{21} + \)\(56\!\cdots\!92\)\( p^{63} T^{22} - \)\(31\!\cdots\!04\)\( p^{70} T^{23} + 323406546094099 p^{77} T^{24} - 9291302 p^{84} T^{25} + p^{91} T^{26} \)
97 \( 1 - 10001852 T + 551581550822996 T^{2} - \)\(38\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(71\!\cdots\!70\)\( p T^{5} + \)\(23\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!08\)\( T^{7} + \)\(31\!\cdots\!27\)\( T^{8} - \)\(74\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!81\)\( T^{10} - \)\(66\!\cdots\!32\)\( T^{11} + \)\(32\!\cdots\!38\)\( T^{12} - \)\(56\!\cdots\!82\)\( T^{13} + \)\(32\!\cdots\!38\)\( p^{7} T^{14} - \)\(66\!\cdots\!32\)\( p^{14} T^{15} + \)\(34\!\cdots\!81\)\( p^{21} T^{16} - \)\(74\!\cdots\!80\)\( p^{28} T^{17} + \)\(31\!\cdots\!27\)\( p^{35} T^{18} - \)\(80\!\cdots\!08\)\( p^{42} T^{19} + \)\(23\!\cdots\!80\)\( p^{49} T^{20} - \)\(71\!\cdots\!70\)\( p^{57} T^{21} + \)\(14\!\cdots\!86\)\( p^{63} T^{22} - \)\(38\!\cdots\!08\)\( p^{70} T^{23} + 551581550822996 p^{77} T^{24} - 10001852 p^{84} T^{25} + p^{91} T^{26} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.87005227067048405349155023885, −3.62744442250063717915336342541, −3.54342324895824396165458645725, −3.52553292236940281300784223832, −3.51708327321286264357112136707, −3.38232308648273880461724115570, −3.25902929607434875731701060843, −3.25052740233407094299766576552, −3.07878366081793291452126047674, −2.72418982127281489097211794682, −2.62864437113062029615022509759, −2.36711558497486924265681966520, −2.22070434964846614755690740934, −2.14546732053670901113708620622, −2.13827213695954792577595187478, −1.71186247415649031714820697881, −1.36723945722206286405726657027, −1.35412508978399793859627500443, −1.27916372064430856025773472755, −1.24170678859237207613617355618, −0.957960980776983754996256738882, −0.831624383573782639258420854093, −0.55235583804325382276868960182, −0.53689266807612189750130633243, −0.085078150271300746480761381589, 0.085078150271300746480761381589, 0.53689266807612189750130633243, 0.55235583804325382276868960182, 0.831624383573782639258420854093, 0.957960980776983754996256738882, 1.24170678859237207613617355618, 1.27916372064430856025773472755, 1.35412508978399793859627500443, 1.36723945722206286405726657027, 1.71186247415649031714820697881, 2.13827213695954792577595187478, 2.14546732053670901113708620622, 2.22070434964846614755690740934, 2.36711558497486924265681966520, 2.62864437113062029615022509759, 2.72418982127281489097211794682, 3.07878366081793291452126047674, 3.25052740233407094299766576552, 3.25902929607434875731701060843, 3.38232308648273880461724115570, 3.51708327321286264357112136707, 3.52553292236940281300784223832, 3.54342324895824396165458645725, 3.62744442250063717915336342541, 3.87005227067048405349155023885

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

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