Properties

Label 18-825e9-1.1-c5e9-0-0
Degree $18$
Conductor $1.770\times 10^{26}$
Sign $1$
Analytic cond. $1.24316\times 10^{19}$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 81·3-s − 58·4-s − 81·6-s − 57·7-s − 128·8-s + 3.64e3·9-s − 1.08e3·11-s + 4.69e3·12-s + 723·13-s − 57·14-s + 1.28e3·16-s − 2.80e3·17-s + 3.64e3·18-s − 1.60e3·19-s + 4.61e3·21-s − 1.08e3·22-s − 2.39e3·23-s + 1.03e4·24-s + 723·26-s − 1.20e5·27-s + 3.30e3·28-s + 5.96e3·29-s + 2.15e4·31-s + 1.44e3·32-s + 8.82e4·33-s − 2.80e3·34-s + ⋯
L(s)  = 1  + 0.176·2-s − 5.19·3-s − 1.81·4-s − 0.918·6-s − 0.439·7-s − 0.707·8-s + 15·9-s − 2.71·11-s + 9.41·12-s + 1.18·13-s − 0.0777·14-s + 1.25·16-s − 2.35·17-s + 2.65·18-s − 1.01·19-s + 2.28·21-s − 0.479·22-s − 0.942·23-s + 3.67·24-s + 0.209·26-s − 31.7·27-s + 0.796·28-s + 1.31·29-s + 4.03·31-s + 0.249·32-s + 14.1·33-s − 0.415·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{18} \cdot 11^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{18} \cdot 11^{9}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(3^{9} \cdot 5^{18} \cdot 11^{9}\)
Sign: $1$
Analytic conductor: \(1.24316\times 10^{19}\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 3^{9} \cdot 5^{18} \cdot 11^{9} ,\ ( \ : [5/2]^{9} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.8400703029\)
\(L(\frac12)\) \(\approx\) \(0.8400703029\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{2} T )^{9} \)
5 \( 1 \)
11 \( ( 1 + p^{2} T )^{9} \)
good2 \( 1 - T + 59 T^{2} + 11 T^{3} + 1001 p T^{4} + 753 p^{3} T^{5} + 813 p^{6} T^{6} + 21731 p^{4} T^{7} + 32045 p^{5} T^{8} + 27195 p^{9} T^{9} + 32045 p^{10} T^{10} + 21731 p^{14} T^{11} + 813 p^{21} T^{12} + 753 p^{23} T^{13} + 1001 p^{26} T^{14} + 11 p^{30} T^{15} + 59 p^{35} T^{16} - p^{40} T^{17} + p^{45} T^{18} \)
7 \( 1 + 57 T + 86990 T^{2} + 4695527 T^{3} + 4003143790 T^{4} + 28895610631 p T^{5} + 123148937158413 T^{6} + 5661740312705338 T^{7} + 2757244372640068232 T^{8} + \)\(11\!\cdots\!18\)\( T^{9} + 2757244372640068232 p^{5} T^{10} + 5661740312705338 p^{10} T^{11} + 123148937158413 p^{15} T^{12} + 28895610631 p^{21} T^{13} + 4003143790 p^{25} T^{14} + 4695527 p^{30} T^{15} + 86990 p^{35} T^{16} + 57 p^{40} T^{17} + p^{45} T^{18} \)
13 \( 1 - 723 T + 1462252 T^{2} - 1143917061 T^{3} + 1336464049441 T^{4} - 983929350911122 T^{5} + 857593698768315423 T^{6} - \)\(57\!\cdots\!63\)\( T^{7} + \)\(41\!\cdots\!07\)\( T^{8} - \)\(24\!\cdots\!06\)\( T^{9} + \)\(41\!\cdots\!07\)\( p^{5} T^{10} - \)\(57\!\cdots\!63\)\( p^{10} T^{11} + 857593698768315423 p^{15} T^{12} - 983929350911122 p^{20} T^{13} + 1336464049441 p^{25} T^{14} - 1143917061 p^{30} T^{15} + 1462252 p^{35} T^{16} - 723 p^{40} T^{17} + p^{45} T^{18} \)
17 \( 1 + 2804 T + 8741378 T^{2} + 12763168902 T^{3} + 22444731823890 T^{4} + 17945122467394308 T^{5} + 23870209519791977019 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(84\!\cdots\!68\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{9} + \)\(84\!\cdots\!68\)\( p^{5} T^{10} + \)\(10\!\cdots\!12\)\( p^{10} T^{11} + 23870209519791977019 p^{15} T^{12} + 17945122467394308 p^{20} T^{13} + 22444731823890 p^{25} T^{14} + 12763168902 p^{30} T^{15} + 8741378 p^{35} T^{16} + 2804 p^{40} T^{17} + p^{45} T^{18} \)
19 \( 1 + 1601 T + 12851473 T^{2} + 19957709258 T^{3} + 77274340162926 T^{4} + 124222587208756568 T^{5} + \)\(30\!\cdots\!56\)\( T^{6} + \)\(50\!\cdots\!72\)\( T^{7} + \)\(91\!\cdots\!53\)\( T^{8} + \)\(14\!\cdots\!25\)\( T^{9} + \)\(91\!\cdots\!53\)\( p^{5} T^{10} + \)\(50\!\cdots\!72\)\( p^{10} T^{11} + \)\(30\!\cdots\!56\)\( p^{15} T^{12} + 124222587208756568 p^{20} T^{13} + 77274340162926 p^{25} T^{14} + 19957709258 p^{30} T^{15} + 12851473 p^{35} T^{16} + 1601 p^{40} T^{17} + p^{45} T^{18} \)
23 \( 1 + 104 p T + 35251685 T^{2} + 92017693262 T^{3} + 663624861666434 T^{4} + 1662308224986931890 T^{5} + \)\(82\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!64\)\( T^{7} + \)\(72\!\cdots\!49\)\( T^{8} + \)\(14\!\cdots\!86\)\( T^{9} + \)\(72\!\cdots\!49\)\( p^{5} T^{10} + \)\(18\!\cdots\!64\)\( p^{10} T^{11} + \)\(82\!\cdots\!28\)\( p^{15} T^{12} + 1662308224986931890 p^{20} T^{13} + 663624861666434 p^{25} T^{14} + 92017693262 p^{30} T^{15} + 35251685 p^{35} T^{16} + 104 p^{41} T^{17} + p^{45} T^{18} \)
29 \( 1 - 5966 T + 50635824 T^{2} - 458178162542 T^{3} + 84376405149577 p T^{4} - 15209438833794638524 T^{5} + \)\(88\!\cdots\!75\)\( T^{6} - \)\(42\!\cdots\!26\)\( T^{7} + \)\(21\!\cdots\!59\)\( T^{8} - \)\(10\!\cdots\!32\)\( T^{9} + \)\(21\!\cdots\!59\)\( p^{5} T^{10} - \)\(42\!\cdots\!26\)\( p^{10} T^{11} + \)\(88\!\cdots\!75\)\( p^{15} T^{12} - 15209438833794638524 p^{20} T^{13} + 84376405149577 p^{26} T^{14} - 458178162542 p^{30} T^{15} + 50635824 p^{35} T^{16} - 5966 p^{40} T^{17} + p^{45} T^{18} \)
31 \( 1 - 21575 T + 299756334 T^{2} - 2640086113929 T^{3} + 18420316113910379 T^{4} - \)\(10\!\cdots\!98\)\( T^{5} + \)\(60\!\cdots\!95\)\( T^{6} - \)\(38\!\cdots\!67\)\( T^{7} + \)\(26\!\cdots\!89\)\( T^{8} - \)\(15\!\cdots\!46\)\( T^{9} + \)\(26\!\cdots\!89\)\( p^{5} T^{10} - \)\(38\!\cdots\!67\)\( p^{10} T^{11} + \)\(60\!\cdots\!95\)\( p^{15} T^{12} - \)\(10\!\cdots\!98\)\( p^{20} T^{13} + 18420316113910379 p^{25} T^{14} - 2640086113929 p^{30} T^{15} + 299756334 p^{35} T^{16} - 21575 p^{40} T^{17} + p^{45} T^{18} \)
37 \( 1 + 10228 T + 342138524 T^{2} + 1773562886782 T^{3} + 43079189695415340 T^{4} + 53010840543137067364 T^{5} + \)\(32\!\cdots\!91\)\( T^{6} - \)\(71\!\cdots\!04\)\( T^{7} + \)\(22\!\cdots\!72\)\( T^{8} - \)\(80\!\cdots\!20\)\( T^{9} + \)\(22\!\cdots\!72\)\( p^{5} T^{10} - \)\(71\!\cdots\!04\)\( p^{10} T^{11} + \)\(32\!\cdots\!91\)\( p^{15} T^{12} + 53010840543137067364 p^{20} T^{13} + 43079189695415340 p^{25} T^{14} + 1773562886782 p^{30} T^{15} + 342138524 p^{35} T^{16} + 10228 p^{40} T^{17} + p^{45} T^{18} \)
41 \( 1 - 13304 T + 774520542 T^{2} - 8961679883518 T^{3} + 286574740069871914 T^{4} - \)\(28\!\cdots\!88\)\( T^{5} + \)\(66\!\cdots\!03\)\( T^{6} - \)\(58\!\cdots\!84\)\( T^{7} + \)\(10\!\cdots\!64\)\( T^{8} - \)\(80\!\cdots\!68\)\( T^{9} + \)\(10\!\cdots\!64\)\( p^{5} T^{10} - \)\(58\!\cdots\!84\)\( p^{10} T^{11} + \)\(66\!\cdots\!03\)\( p^{15} T^{12} - \)\(28\!\cdots\!88\)\( p^{20} T^{13} + 286574740069871914 p^{25} T^{14} - 8961679883518 p^{30} T^{15} + 774520542 p^{35} T^{16} - 13304 p^{40} T^{17} + p^{45} T^{18} \)
43 \( 1 + 13829 T + 1007459350 T^{2} + 12514774475459 T^{3} + 485496776664396771 T^{4} + \)\(53\!\cdots\!78\)\( T^{5} + \)\(14\!\cdots\!99\)\( T^{6} + \)\(14\!\cdots\!13\)\( T^{7} + \)\(30\!\cdots\!29\)\( T^{8} + \)\(24\!\cdots\!30\)\( T^{9} + \)\(30\!\cdots\!29\)\( p^{5} T^{10} + \)\(14\!\cdots\!13\)\( p^{10} T^{11} + \)\(14\!\cdots\!99\)\( p^{15} T^{12} + \)\(53\!\cdots\!78\)\( p^{20} T^{13} + 485496776664396771 p^{25} T^{14} + 12514774475459 p^{30} T^{15} + 1007459350 p^{35} T^{16} + 13829 p^{40} T^{17} + p^{45} T^{18} \)
47 \( 1 + 13998 T + 1205232710 T^{2} + 13274958912068 T^{3} + 719508279987808858 T^{4} + \)\(65\!\cdots\!90\)\( T^{5} + \)\(28\!\cdots\!13\)\( T^{6} + \)\(21\!\cdots\!48\)\( T^{7} + \)\(83\!\cdots\!76\)\( T^{8} + \)\(56\!\cdots\!64\)\( T^{9} + \)\(83\!\cdots\!76\)\( p^{5} T^{10} + \)\(21\!\cdots\!48\)\( p^{10} T^{11} + \)\(28\!\cdots\!13\)\( p^{15} T^{12} + \)\(65\!\cdots\!90\)\( p^{20} T^{13} + 719508279987808858 p^{25} T^{14} + 13274958912068 p^{30} T^{15} + 1205232710 p^{35} T^{16} + 13998 p^{40} T^{17} + p^{45} T^{18} \)
53 \( 1 + 44166 T + 2964779929 T^{2} + 87188141140712 T^{3} + 3403427266005613816 T^{4} + \)\(73\!\cdots\!52\)\( T^{5} + \)\(21\!\cdots\!32\)\( T^{6} + \)\(37\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!34\)\( T^{8} + \)\(15\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!34\)\( p^{5} T^{10} + \)\(37\!\cdots\!40\)\( p^{10} T^{11} + \)\(21\!\cdots\!32\)\( p^{15} T^{12} + \)\(73\!\cdots\!52\)\( p^{20} T^{13} + 3403427266005613816 p^{25} T^{14} + 87188141140712 p^{30} T^{15} + 2964779929 p^{35} T^{16} + 44166 p^{40} T^{17} + p^{45} T^{18} \)
59 \( 1 - 11626 T + 2850942718 T^{2} - 33809265359684 T^{3} + 4084641425986968422 T^{4} - \)\(40\!\cdots\!22\)\( T^{5} + \)\(38\!\cdots\!09\)\( T^{6} - \)\(29\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!40\)\( T^{8} - \)\(19\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!40\)\( p^{5} T^{10} - \)\(29\!\cdots\!12\)\( p^{10} T^{11} + \)\(38\!\cdots\!09\)\( p^{15} T^{12} - \)\(40\!\cdots\!22\)\( p^{20} T^{13} + 4084641425986968422 p^{25} T^{14} - 33809265359684 p^{30} T^{15} + 2850942718 p^{35} T^{16} - 11626 p^{40} T^{17} + p^{45} T^{18} \)
61 \( 1 - 49481 T + 4609993897 T^{2} - 171638702043048 T^{3} + 8883884187287210928 T^{4} - \)\(29\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} - \)\(38\!\cdots\!88\)\( T^{7} + \)\(12\!\cdots\!18\)\( T^{8} - \)\(38\!\cdots\!30\)\( T^{9} + \)\(12\!\cdots\!18\)\( p^{5} T^{10} - \)\(38\!\cdots\!88\)\( p^{10} T^{11} + \)\(11\!\cdots\!68\)\( p^{15} T^{12} - \)\(29\!\cdots\!80\)\( p^{20} T^{13} + 8883884187287210928 p^{25} T^{14} - 171638702043048 p^{30} T^{15} + 4609993897 p^{35} T^{16} - 49481 p^{40} T^{17} + p^{45} T^{18} \)
67 \( 1 - 26567 T + 6847580423 T^{2} - 176605220883648 T^{3} + 23485476893685848128 T^{4} - \)\(57\!\cdots\!48\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} - \)\(12\!\cdots\!28\)\( T^{7} + \)\(97\!\cdots\!06\)\( T^{8} - \)\(19\!\cdots\!02\)\( T^{9} + \)\(97\!\cdots\!06\)\( p^{5} T^{10} - \)\(12\!\cdots\!28\)\( p^{10} T^{11} + \)\(55\!\cdots\!60\)\( p^{15} T^{12} - \)\(57\!\cdots\!48\)\( p^{20} T^{13} + 23485476893685848128 p^{25} T^{14} - 176605220883648 p^{30} T^{15} + 6847580423 p^{35} T^{16} - 26567 p^{40} T^{17} + p^{45} T^{18} \)
71 \( 1 - 78454 T + 9994813401 T^{2} - 661066751932506 T^{3} + 52322480337735347866 T^{4} - \)\(28\!\cdots\!22\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(82\!\cdots\!52\)\( T^{7} + \)\(42\!\cdots\!17\)\( T^{8} - \)\(17\!\cdots\!36\)\( T^{9} + \)\(42\!\cdots\!17\)\( p^{5} T^{10} - \)\(82\!\cdots\!52\)\( p^{10} T^{11} + \)\(17\!\cdots\!28\)\( p^{15} T^{12} - \)\(28\!\cdots\!22\)\( p^{20} T^{13} + 52322480337735347866 p^{25} T^{14} - 661066751932506 p^{30} T^{15} + 9994813401 p^{35} T^{16} - 78454 p^{40} T^{17} + p^{45} T^{18} \)
73 \( 1 + 100086 T + 17738649209 T^{2} + 1276648484234752 T^{3} + \)\(12\!\cdots\!48\)\( T^{4} + \)\(73\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!28\)\( T^{7} + \)\(15\!\cdots\!02\)\( T^{8} + \)\(62\!\cdots\!32\)\( T^{9} + \)\(15\!\cdots\!02\)\( p^{5} T^{10} + \)\(25\!\cdots\!28\)\( p^{10} T^{11} + \)\(54\!\cdots\!60\)\( p^{15} T^{12} + \)\(73\!\cdots\!80\)\( p^{20} T^{13} + \)\(12\!\cdots\!48\)\( p^{25} T^{14} + 1276648484234752 p^{30} T^{15} + 17738649209 p^{35} T^{16} + 100086 p^{40} T^{17} + p^{45} T^{18} \)
79 \( 1 + 8478 T + 12280336668 T^{2} + 26375638319276 T^{3} + 83075363415546786864 T^{4} + \)\(36\!\cdots\!74\)\( T^{5} + \)\(41\!\cdots\!29\)\( T^{6} - \)\(73\!\cdots\!04\)\( T^{7} + \)\(15\!\cdots\!96\)\( T^{8} - \)\(86\!\cdots\!04\)\( T^{9} + \)\(15\!\cdots\!96\)\( p^{5} T^{10} - \)\(73\!\cdots\!04\)\( p^{10} T^{11} + \)\(41\!\cdots\!29\)\( p^{15} T^{12} + \)\(36\!\cdots\!74\)\( p^{20} T^{13} + 83075363415546786864 p^{25} T^{14} + 26375638319276 p^{30} T^{15} + 12280336668 p^{35} T^{16} + 8478 p^{40} T^{17} + p^{45} T^{18} \)
83 \( 1 + 157476 T + 29773766160 T^{2} + 3204691473919578 T^{3} + \)\(38\!\cdots\!89\)\( T^{4} + \)\(32\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!65\)\( T^{6} + \)\(21\!\cdots\!42\)\( T^{7} + \)\(16\!\cdots\!91\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(16\!\cdots\!91\)\( p^{5} T^{10} + \)\(21\!\cdots\!42\)\( p^{10} T^{11} + \)\(30\!\cdots\!65\)\( p^{15} T^{12} + \)\(32\!\cdots\!08\)\( p^{20} T^{13} + \)\(38\!\cdots\!89\)\( p^{25} T^{14} + 3204691473919578 p^{30} T^{15} + 29773766160 p^{35} T^{16} + 157476 p^{40} T^{17} + p^{45} T^{18} \)
89 \( 1 + 65548 T + 34227705138 T^{2} + 2624626620125446 T^{3} + \)\(55\!\cdots\!55\)\( T^{4} + \)\(46\!\cdots\!60\)\( T^{5} + \)\(56\!\cdots\!81\)\( T^{6} + \)\(48\!\cdots\!90\)\( T^{7} + \)\(41\!\cdots\!81\)\( T^{8} + \)\(33\!\cdots\!68\)\( T^{9} + \)\(41\!\cdots\!81\)\( p^{5} T^{10} + \)\(48\!\cdots\!90\)\( p^{10} T^{11} + \)\(56\!\cdots\!81\)\( p^{15} T^{12} + \)\(46\!\cdots\!60\)\( p^{20} T^{13} + \)\(55\!\cdots\!55\)\( p^{25} T^{14} + 2624626620125446 p^{30} T^{15} + 34227705138 p^{35} T^{16} + 65548 p^{40} T^{17} + p^{45} T^{18} \)
97 \( 1 - 116757 T + 43988130361 T^{2} - 5079516869456032 T^{3} + \)\(93\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!18\)\( T^{5} + \)\(13\!\cdots\!30\)\( T^{6} - \)\(13\!\cdots\!76\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(12\!\cdots\!63\)\( T^{9} + \)\(15\!\cdots\!87\)\( p^{5} T^{10} - \)\(13\!\cdots\!76\)\( p^{10} T^{11} + \)\(13\!\cdots\!30\)\( p^{15} T^{12} - \)\(10\!\cdots\!18\)\( p^{20} T^{13} + \)\(93\!\cdots\!38\)\( p^{25} T^{14} - 5079516869456032 p^{30} T^{15} + 43988130361 p^{35} T^{16} - 116757 p^{40} T^{17} + p^{45} T^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.21204433641827806836887775173, −3.19864886352344614485801750953, −3.14449623359567758439067793339, −3.01005132576675512610223140724, −2.91286078916180609484240558203, −2.52040986420128791001428398857, −2.34740040754293296222443322203, −2.32436192962060808319426831318, −2.22084485196809432002511854716, −2.15634080463039490484642105593, −1.77408696669920202968660601991, −1.64815187419321730457328856184, −1.64258465624372901292740152231, −1.52989506334743613380374036920, −1.41942410520883605914599261814, −1.33508644169725273204226593857, −1.02601573771067402209216837316, −0.75955284656520670844282224019, −0.60143291256109746847266096642, −0.51591478659613766708256093230, −0.49786675485691329818516983671, −0.42853353168514640284742684831, −0.38285967482634164063268485726, −0.28261528811720607172130212163, −0.21652399922255474231109422067, 0.21652399922255474231109422067, 0.28261528811720607172130212163, 0.38285967482634164063268485726, 0.42853353168514640284742684831, 0.49786675485691329818516983671, 0.51591478659613766708256093230, 0.60143291256109746847266096642, 0.75955284656520670844282224019, 1.02601573771067402209216837316, 1.33508644169725273204226593857, 1.41942410520883605914599261814, 1.52989506334743613380374036920, 1.64258465624372901292740152231, 1.64815187419321730457328856184, 1.77408696669920202968660601991, 2.15634080463039490484642105593, 2.22084485196809432002511854716, 2.32436192962060808319426831318, 2.34740040754293296222443322203, 2.52040986420128791001428398857, 2.91286078916180609484240558203, 3.01005132576675512610223140724, 3.14449623359567758439067793339, 3.19864886352344614485801750953, 3.21204433641827806836887775173

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

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