Properties

Degree 20
Conductor $ 2^{20} \cdot 3^{30} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 21·5-s + 29·7-s − 177·11-s − 181·13-s − 2.28e3·17-s − 832·19-s − 399·23-s + 5.64e3·25-s + 6.03e3·29-s + 2.75e3·31-s + 609·35-s − 1.51e4·37-s + 1.84e4·41-s + 1.46e3·43-s + 2.51e4·47-s + 4.04e4·49-s − 1.16e5·53-s − 3.71e3·55-s + 9.05e4·59-s + 1.40e3·61-s − 3.80e3·65-s + 1.39e4·67-s − 2.29e5·71-s + 1.52e4·73-s − 5.13e3·77-s + 2.99e4·79-s + 2.28e5·83-s + ⋯
L(s)  = 1  + 0.375·5-s + 0.223·7-s − 0.441·11-s − 0.297·13-s − 1.91·17-s − 0.528·19-s − 0.157·23-s + 1.80·25-s + 1.33·29-s + 0.515·31-s + 0.0840·35-s − 1.82·37-s + 1.71·41-s + 0.121·43-s + 1.66·47-s + 2.40·49-s − 5.71·53-s − 0.165·55-s + 3.38·59-s + 0.0482·61-s − 0.111·65-s + 0.378·67-s − 5.39·71-s + 0.333·73-s − 0.0986·77-s + 0.540·79-s + 3.64·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{20} \cdot 3^{30}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((20,\ 2^{20} \cdot 3^{30} ,\ ( \ : [5/2]^{10} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(5.58646\)
\(L(\frac12)\)  \(\approx\)  \(5.58646\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 20. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 21 T - 5203 T^{2} + 103986 p T^{3} + 14035794 T^{4} - 570964554 p T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 476972383330224 p^{2} T^{9} + 2533145723872694124 T^{10} - 476972383330224 p^{7} T^{11} - 599011867854189 p^{10} T^{12} + 9761967315441 p^{15} T^{13} + 76722872007 p^{20} T^{14} - 570964554 p^{26} T^{15} + 14035794 p^{30} T^{16} + 103986 p^{36} T^{17} - 5203 p^{40} T^{18} - 21 p^{45} T^{19} + p^{50} T^{20} \)
7 \( 1 - 29 T - 39569 T^{2} + 537492 p T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 395584797729 p T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} - \)\(69\!\cdots\!60\)\( T^{10} + 16031898530170676332 p^{5} T^{11} + 238615372451780007 p^{10} T^{12} - 560172784984473 p^{15} T^{13} - 395584797729 p^{21} T^{14} - 77352503496 p^{25} T^{15} + 440397336 p^{30} T^{16} + 537492 p^{36} T^{17} - 39569 p^{40} T^{18} - 29 p^{45} T^{19} + p^{50} T^{20} \)
11 \( 1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} - \)\(10\!\cdots\!51\)\( T^{8} - \)\(50\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} - \)\(50\!\cdots\!82\)\( p^{5} T^{11} - \)\(10\!\cdots\!51\)\( p^{10} T^{12} + 8552599160812456257 p^{15} T^{13} - 6913987980717606 p^{20} T^{14} - 33726096455301 p^{25} T^{15} + 104816625882 p^{30} T^{16} + 71434269 p^{35} T^{17} - 396232 p^{40} T^{18} + 177 p^{45} T^{19} + p^{50} T^{20} \)
13 \( 1 + 181 T - 1012331 T^{2} + 1114014 p T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 14806115812129521 p T^{6} - 802846347498861897 T^{7} + \)\(98\!\cdots\!51\)\( T^{8} + \)\(77\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!24\)\( T^{10} + \)\(77\!\cdots\!72\)\( p^{5} T^{11} + \)\(98\!\cdots\!51\)\( p^{10} T^{12} - 802846347498861897 p^{15} T^{13} - 14806115812129521 p^{21} T^{14} - 84043375137762 p^{25} T^{15} + 446454243174 p^{30} T^{16} + 1114014 p^{36} T^{17} - 1012331 p^{40} T^{18} + 181 p^{45} T^{19} + p^{50} T^{20} \)
17 \( ( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 10068870522169 p^{5} T^{6} + 3443850354 p^{10} T^{7} + 4980550 p^{15} T^{8} + 1140 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
19 \( ( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 20272296121125 p^{5} T^{6} + 6215761044 p^{10} T^{7} + 5046258 p^{15} T^{8} + 416 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
23 \( 1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} + \)\(48\!\cdots\!51\)\( T^{7} - \)\(71\!\cdots\!41\)\( T^{8} - \)\(50\!\cdots\!24\)\( p T^{9} + \)\(81\!\cdots\!80\)\( T^{10} - \)\(50\!\cdots\!24\)\( p^{6} T^{11} - \)\(71\!\cdots\!41\)\( p^{10} T^{12} + \)\(48\!\cdots\!51\)\( p^{15} T^{13} - 48522958516353490863 p^{20} T^{14} - 562944417983120520 p^{25} T^{15} + 155650662506976 p^{30} T^{16} + 38108825820 p^{35} T^{17} - 16077241 p^{40} T^{18} + 399 p^{45} T^{19} + p^{50} T^{20} \)
29 \( 1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} + \)\(64\!\cdots\!39\)\( T^{6} - \)\(82\!\cdots\!55\)\( T^{7} - \)\(90\!\cdots\!17\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!16\)\( T^{10} - \)\(14\!\cdots\!60\)\( p^{5} T^{11} - \)\(90\!\cdots\!17\)\( p^{10} T^{12} - \)\(82\!\cdots\!55\)\( p^{15} T^{13} + \)\(64\!\cdots\!39\)\( p^{20} T^{14} - 668469168127712358 p^{25} T^{15} + 283528398607854 p^{30} T^{16} - 31641196734 p^{35} T^{17} + 3652157 p^{40} T^{18} - 6033 p^{45} T^{19} + p^{50} T^{20} \)
31 \( 1 - 89 p T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} - \)\(32\!\cdots\!47\)\( T^{6} - \)\(42\!\cdots\!39\)\( T^{7} - \)\(13\!\cdots\!49\)\( T^{8} + \)\(36\!\cdots\!48\)\( T^{9} + \)\(63\!\cdots\!24\)\( T^{10} + \)\(36\!\cdots\!48\)\( p^{5} T^{11} - \)\(13\!\cdots\!49\)\( p^{10} T^{12} - \)\(42\!\cdots\!39\)\( p^{15} T^{13} - \)\(32\!\cdots\!47\)\( p^{20} T^{14} + 13274031992302596720 p^{25} T^{15} + 2052158291804100 p^{30} T^{16} - 189444651072 p^{35} T^{17} - 54902477 p^{40} T^{18} - 89 p^{46} T^{19} + p^{50} T^{20} \)
37 \( ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + 19241810738464926 p^{5} T^{6} + 803146672896 p^{10} T^{7} + 201201093 p^{15} T^{8} + 7586 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
41 \( 1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} - \)\(52\!\cdots\!97\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!43\)\( T^{7} - \)\(76\!\cdots\!83\)\( p T^{8} - \)\(10\!\cdots\!42\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} - \)\(10\!\cdots\!42\)\( p^{5} T^{11} - \)\(76\!\cdots\!83\)\( p^{11} T^{12} + \)\(39\!\cdots\!43\)\( p^{15} T^{13} + \)\(16\!\cdots\!40\)\( p^{20} T^{14} - \)\(52\!\cdots\!97\)\( p^{25} T^{15} - 505249106564622 p^{30} T^{16} + 4344492069675 p^{35} T^{17} - 117679042 p^{40} T^{18} - 18435 p^{45} T^{19} + p^{50} T^{20} \)
43 \( 1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} - \)\(75\!\cdots\!27\)\( T^{5} + \)\(55\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!57\)\( T^{7} - \)\(29\!\cdots\!39\)\( T^{8} + \)\(61\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!84\)\( T^{10} + \)\(61\!\cdots\!62\)\( p^{5} T^{11} - \)\(29\!\cdots\!39\)\( p^{10} T^{12} - \)\(12\!\cdots\!57\)\( p^{15} T^{13} + \)\(55\!\cdots\!62\)\( p^{20} T^{14} - \)\(75\!\cdots\!27\)\( p^{25} T^{15} + 12129147672135834 p^{30} T^{16} + 4016430594327 p^{35} T^{17} - 271863536 p^{40} T^{18} - 1469 p^{45} T^{19} + p^{50} T^{20} \)
47 \( 1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} - \)\(41\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!27\)\( T^{6} + \)\(55\!\cdots\!85\)\( T^{7} + \)\(12\!\cdots\!11\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{9} - \)\(37\!\cdots\!60\)\( T^{10} - \)\(42\!\cdots\!56\)\( p^{5} T^{11} + \)\(12\!\cdots\!11\)\( p^{10} T^{12} + \)\(55\!\cdots\!85\)\( p^{15} T^{13} - \)\(25\!\cdots\!27\)\( p^{20} T^{14} - \)\(41\!\cdots\!12\)\( p^{25} T^{15} + 97557609874842960 p^{30} T^{16} + 14349179861244 p^{35} T^{17} - 401246233 p^{40} T^{18} - 25155 p^{45} T^{19} + p^{50} T^{20} \)
53 \( ( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + 3390725554692289246 p^{5} T^{6} + 110313236959296 p^{10} T^{7} + 3354568213 p^{15} T^{8} + 58422 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
59 \( 1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} + \)\(10\!\cdots\!09\)\( T^{5} + \)\(69\!\cdots\!78\)\( T^{6} - \)\(67\!\cdots\!27\)\( p T^{7} + \)\(84\!\cdots\!01\)\( T^{8} + \)\(17\!\cdots\!74\)\( T^{9} - \)\(12\!\cdots\!20\)\( T^{10} + \)\(17\!\cdots\!74\)\( p^{5} T^{11} + \)\(84\!\cdots\!01\)\( p^{10} T^{12} - \)\(67\!\cdots\!27\)\( p^{16} T^{13} + \)\(69\!\cdots\!78\)\( p^{20} T^{14} + \)\(10\!\cdots\!09\)\( p^{25} T^{15} - 966660594685472478 p^{30} T^{16} - 13805150996349 p^{35} T^{17} + 2831117840 p^{40} T^{18} - 90537 p^{45} T^{19} + p^{50} T^{20} \)
61 \( 1 - 23 p T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(10\!\cdots\!33\)\( T^{6} - \)\(70\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!67\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!04\)\( p^{5} T^{11} + \)\(11\!\cdots\!67\)\( p^{10} T^{12} - \)\(70\!\cdots\!89\)\( p^{15} T^{13} - \)\(10\!\cdots\!33\)\( p^{20} T^{14} + \)\(54\!\cdots\!90\)\( p^{25} T^{15} + 7065863261737144698 p^{30} T^{16} - 452840008146 p^{35} T^{17} - 3536905883 p^{40} T^{18} - 23 p^{46} T^{19} + p^{50} T^{20} \)
67 \( 1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} + \)\(30\!\cdots\!39\)\( T^{5} - \)\(79\!\cdots\!54\)\( T^{6} - \)\(69\!\cdots\!51\)\( T^{7} - \)\(33\!\cdots\!67\)\( T^{8} + \)\(37\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!76\)\( T^{10} + \)\(37\!\cdots\!46\)\( p^{5} T^{11} - \)\(33\!\cdots\!67\)\( p^{10} T^{12} - \)\(69\!\cdots\!51\)\( p^{15} T^{13} - \)\(79\!\cdots\!54\)\( p^{20} T^{14} + \)\(30\!\cdots\!39\)\( p^{25} T^{15} + 10014688417385231130 p^{30} T^{16} - 77425491657903 p^{35} T^{17} - 3876685544 p^{40} T^{18} - 13907 p^{45} T^{19} + p^{50} T^{20} \)
71 \( ( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} + \)\(77\!\cdots\!40\)\( T^{5} + 19260501229393543450 p^{5} T^{6} + 426246123888336 p^{10} T^{7} + 7758380659 p^{15} T^{8} + 114684 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
73 \( ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + 12288417972789256281 p^{5} T^{6} - 31056473559714 p^{10} T^{7} + 3606834246 p^{15} T^{8} - 7600 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
79 \( 1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!85\)\( T^{6} - \)\(84\!\cdots\!45\)\( T^{7} - \)\(35\!\cdots\!45\)\( T^{8} + \)\(81\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} + \)\(81\!\cdots\!80\)\( p^{5} T^{11} - \)\(35\!\cdots\!45\)\( p^{10} T^{12} - \)\(84\!\cdots\!45\)\( p^{15} T^{13} + \)\(27\!\cdots\!85\)\( p^{20} T^{14} + \)\(21\!\cdots\!52\)\( p^{25} T^{15} + 26234825811851125236 p^{30} T^{16} - 358913063028768 p^{35} T^{17} - 5352351629 p^{40} T^{18} - 29993 p^{45} T^{19} + p^{50} T^{20} \)
83 \( 1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} - \)\(36\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!01\)\( T^{6} + \)\(11\!\cdots\!41\)\( T^{7} - \)\(18\!\cdots\!73\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} - \)\(88\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!84\)\( p^{5} T^{11} - \)\(18\!\cdots\!73\)\( p^{10} T^{12} + \)\(11\!\cdots\!41\)\( p^{15} T^{13} + \)\(11\!\cdots\!01\)\( p^{20} T^{14} - \)\(36\!\cdots\!68\)\( p^{25} T^{15} + 62567029919071222368 p^{30} T^{16} - 1202282302650156 p^{35} T^{17} + 21403431983 p^{40} T^{18} - 228951 p^{45} T^{19} + p^{50} T^{20} \)
89 \( ( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(67\!\cdots\!10\)\( p^{5} T^{6} + 6660261403977288 p^{10} T^{7} + 52616244181 p^{15} T^{8} + 299166 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
97 \( 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - \)\(30\!\cdots\!95\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{7} - \)\(20\!\cdots\!15\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(65\!\cdots\!06\)\( p^{5} T^{11} - \)\(20\!\cdots\!15\)\( p^{10} T^{12} + \)\(21\!\cdots\!29\)\( p^{15} T^{13} + \)\(10\!\cdots\!20\)\( p^{20} T^{14} - \)\(30\!\cdots\!95\)\( p^{25} T^{15} + 99710157551726941410 p^{30} T^{16} + 2263333692661293 p^{35} T^{17} - 17893496138 p^{40} T^{18} - 40541 p^{45} T^{19} + p^{50} T^{20} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.40736139761940738563755020042, −4.36090043564385087754783929566, −4.22142387235296441314776340950, −4.17402897520050080214596573639, −3.96878384367455217808690191139, −3.59087565741181944355999668479, −3.57100555346796175631603396324, −3.24600332897800041844240350576, −3.13950726285131638840487566065, −2.96648729107531819423617426535, −2.80808368853471855466939628982, −2.78668236542198874491203066340, −2.47599770210853296893627410271, −2.35762465786687510262631937692, −2.24116508133253446554311065515, −1.98470357542313506863947734377, −1.72694622891806187122940312168, −1.64418581530981571673711098479, −1.33267262102304111631616495452, −1.33096837679867960779078959382, −0.925270396552731578992928963350, −0.69289580725881483841192344883, −0.53550825493019826634703047970, −0.37801632988307379797940395377, −0.17538671459731487170636270993, 0.17538671459731487170636270993, 0.37801632988307379797940395377, 0.53550825493019826634703047970, 0.69289580725881483841192344883, 0.925270396552731578992928963350, 1.33096837679867960779078959382, 1.33267262102304111631616495452, 1.64418581530981571673711098479, 1.72694622891806187122940312168, 1.98470357542313506863947734377, 2.24116508133253446554311065515, 2.35762465786687510262631937692, 2.47599770210853296893627410271, 2.78668236542198874491203066340, 2.80808368853471855466939628982, 2.96648729107531819423617426535, 3.13950726285131638840487566065, 3.24600332897800041844240350576, 3.57100555346796175631603396324, 3.59087565741181944355999668479, 3.96878384367455217808690191139, 4.17402897520050080214596573639, 4.22142387235296441314776340950, 4.36090043564385087754783929566, 4.40736139761940738563755020042

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

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