Properties

Label 16-825e8-1.1-c5e8-0-1
Degree $16$
Conductor $2.146\times 10^{23}$
Sign $1$
Analytic cond. $9.39541\times 10^{16}$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s + 72·3-s − 34·4-s − 648·6-s + 66·7-s + 576·8-s + 2.91e3·9-s − 968·11-s − 2.44e3·12-s − 382·13-s − 594·14-s − 1.14e3·16-s − 288·17-s − 2.62e4·18-s − 988·19-s + 4.75e3·21-s + 8.71e3·22-s − 5.97e3·23-s + 4.14e4·24-s + 3.43e3·26-s + 8.74e4·27-s − 2.24e3·28-s + 1.03e3·29-s − 4.68e3·31-s − 8.34e3·32-s − 6.96e4·33-s + 2.59e3·34-s + ⋯
L(s)  = 1  − 1.59·2-s + 4.61·3-s − 1.06·4-s − 7.34·6-s + 0.509·7-s + 3.18·8-s + 12·9-s − 2.41·11-s − 4.90·12-s − 0.626·13-s − 0.809·14-s − 1.12·16-s − 0.241·17-s − 19.0·18-s − 0.627·19-s + 2.35·21-s + 3.83·22-s − 2.35·23-s + 14.6·24-s + 0.997·26-s + 23.0·27-s − 0.540·28-s + 0.227·29-s − 0.875·31-s − 1.44·32-s − 11.1·33-s + 0.384·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(9.39541\times 10^{16}\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{2} T )^{8} \)
5 \( 1 \)
11 \( ( 1 + p^{2} T )^{8} \)
good2 \( 1 + 9 T + 115 T^{2} + 765 T^{3} + 845 p^{3} T^{4} + 19649 p T^{5} + 76865 p^{2} T^{6} + 201789 p^{3} T^{7} + 687203 p^{4} T^{8} + 201789 p^{8} T^{9} + 76865 p^{12} T^{10} + 19649 p^{16} T^{11} + 845 p^{23} T^{12} + 765 p^{25} T^{13} + 115 p^{30} T^{14} + 9 p^{35} T^{15} + p^{40} T^{16} \)
7 \( 1 - 66 T + 91139 T^{2} - 4771106 T^{3} + 3771081445 T^{4} - 161361060468 T^{5} + 98229992307334 T^{6} - 3591719419739344 T^{7} + 1877203896962388854 T^{8} - 3591719419739344 p^{5} T^{9} + 98229992307334 p^{10} T^{10} - 161361060468 p^{15} T^{11} + 3771081445 p^{20} T^{12} - 4771106 p^{25} T^{13} + 91139 p^{30} T^{14} - 66 p^{35} T^{15} + p^{40} T^{16} \)
13 \( 1 + 382 T + 1736684 T^{2} + 783427492 T^{3} + 1521185873215 T^{4} + 697738539551720 T^{5} + 910167912756693088 T^{6} + \)\(37\!\cdots\!78\)\( T^{7} + \)\(39\!\cdots\!64\)\( T^{8} + \)\(37\!\cdots\!78\)\( p^{5} T^{9} + 910167912756693088 p^{10} T^{10} + 697738539551720 p^{15} T^{11} + 1521185873215 p^{20} T^{12} + 783427492 p^{25} T^{13} + 1736684 p^{30} T^{14} + 382 p^{35} T^{15} + p^{40} T^{16} \)
17 \( 1 + 288 T + 6322210 T^{2} + 3619142824 T^{3} + 20993085706729 T^{4} + 14242701362198736 T^{5} + 47547004006876580182 T^{6} + \)\(31\!\cdots\!64\)\( T^{7} + \)\(78\!\cdots\!64\)\( T^{8} + \)\(31\!\cdots\!64\)\( p^{5} T^{9} + 47547004006876580182 p^{10} T^{10} + 14242701362198736 p^{15} T^{11} + 20993085706729 p^{20} T^{12} + 3619142824 p^{25} T^{13} + 6322210 p^{30} T^{14} + 288 p^{35} T^{15} + p^{40} T^{16} \)
19 \( 1 + 52 p T + 5959708 T^{2} + 5254912560 T^{3} + 24058425117050 T^{4} + 35113408638958764 T^{5} + 84715734732980301328 T^{6} + \)\(11\!\cdots\!52\)\( T^{7} + \)\(20\!\cdots\!03\)\( T^{8} + \)\(11\!\cdots\!52\)\( p^{5} T^{9} + 84715734732980301328 p^{10} T^{10} + 35113408638958764 p^{15} T^{11} + 24058425117050 p^{20} T^{12} + 5254912560 p^{25} T^{13} + 5959708 p^{30} T^{14} + 52 p^{36} T^{15} + p^{40} T^{16} \)
23 \( 1 + 5972 T + 42122005 T^{2} + 159625929312 T^{3} + 624177713014597 T^{4} + 1641687150293241932 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(97\!\cdots\!60\)\( T^{7} + \)\(27\!\cdots\!54\)\( T^{8} + \)\(97\!\cdots\!60\)\( p^{5} T^{9} + \)\(47\!\cdots\!70\)\( p^{10} T^{10} + 1641687150293241932 p^{15} T^{11} + 624177713014597 p^{20} T^{12} + 159625929312 p^{25} T^{13} + 42122005 p^{30} T^{14} + 5972 p^{35} T^{15} + p^{40} T^{16} \)
29 \( 1 - 1032 T + 83813375 T^{2} - 11786745748 T^{3} + 3065461151481218 T^{4} + 127506026955771500 p T^{5} + \)\(68\!\cdots\!09\)\( T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!78\)\( T^{8} + \)\(18\!\cdots\!84\)\( p^{5} T^{9} + \)\(68\!\cdots\!09\)\( p^{10} T^{10} + 127506026955771500 p^{16} T^{11} + 3065461151481218 p^{20} T^{12} - 11786745748 p^{25} T^{13} + 83813375 p^{30} T^{14} - 1032 p^{35} T^{15} + p^{40} T^{16} \)
31 \( 1 + 4682 T + 150951300 T^{2} + 657057123144 T^{3} + 11207508171334879 T^{4} + 44203316458666562564 T^{5} + \)\(54\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!46\)\( T^{7} + \)\(18\!\cdots\!76\)\( T^{8} + \)\(18\!\cdots\!46\)\( p^{5} T^{9} + \)\(54\!\cdots\!68\)\( p^{10} T^{10} + 44203316458666562564 p^{15} T^{11} + 11207508171334879 p^{20} T^{12} + 657057123144 p^{25} T^{13} + 150951300 p^{30} T^{14} + 4682 p^{35} T^{15} + p^{40} T^{16} \)
37 \( 1 - 17200 T + 324566710 T^{2} - 4365298835948 T^{3} + 58828449991718297 T^{4} - \)\(63\!\cdots\!96\)\( T^{5} + \)\(69\!\cdots\!26\)\( T^{6} - \)\(62\!\cdots\!80\)\( T^{7} + \)\(55\!\cdots\!52\)\( T^{8} - \)\(62\!\cdots\!80\)\( p^{5} T^{9} + \)\(69\!\cdots\!26\)\( p^{10} T^{10} - \)\(63\!\cdots\!96\)\( p^{15} T^{11} + 58828449991718297 p^{20} T^{12} - 4365298835948 p^{25} T^{13} + 324566710 p^{30} T^{14} - 17200 p^{35} T^{15} + p^{40} T^{16} \)
41 \( 1 + 13220 T + 555884974 T^{2} + 5187844762060 T^{3} + 129491274020930057 T^{4} + \)\(85\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!58\)\( T^{6} + \)\(87\!\cdots\!96\)\( T^{7} + \)\(20\!\cdots\!52\)\( T^{8} + \)\(87\!\cdots\!96\)\( p^{5} T^{9} + \)\(17\!\cdots\!58\)\( p^{10} T^{10} + \)\(85\!\cdots\!96\)\( p^{15} T^{11} + 129491274020930057 p^{20} T^{12} + 5187844762060 p^{25} T^{13} + 555884974 p^{30} T^{14} + 13220 p^{35} T^{15} + p^{40} T^{16} \)
43 \( 1 + 22872 T + 679666126 T^{2} + 12782109184832 T^{3} + 228475137913008017 T^{4} + \)\(33\!\cdots\!16\)\( T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(59\!\cdots\!48\)\( T^{7} + \)\(75\!\cdots\!88\)\( T^{8} + \)\(59\!\cdots\!48\)\( p^{5} T^{9} + \)\(47\!\cdots\!70\)\( p^{10} T^{10} + \)\(33\!\cdots\!16\)\( p^{15} T^{11} + 228475137913008017 p^{20} T^{12} + 12782109184832 p^{25} T^{13} + 679666126 p^{30} T^{14} + 22872 p^{35} T^{15} + p^{40} T^{16} \)
47 \( 1 + 6700 T + 733454934 T^{2} + 7406404346624 T^{3} + 284982370032782849 T^{4} + \)\(41\!\cdots\!28\)\( T^{5} + \)\(81\!\cdots\!74\)\( T^{6} + \)\(14\!\cdots\!08\)\( T^{7} + \)\(19\!\cdots\!56\)\( T^{8} + \)\(14\!\cdots\!08\)\( p^{5} T^{9} + \)\(81\!\cdots\!74\)\( p^{10} T^{10} + \)\(41\!\cdots\!28\)\( p^{15} T^{11} + 284982370032782849 p^{20} T^{12} + 7406404346624 p^{25} T^{13} + 733454934 p^{30} T^{14} + 6700 p^{35} T^{15} + p^{40} T^{16} \)
53 \( 1 + 6224 T + 1555305004 T^{2} - 9428770082944 T^{3} + 1170969803856514292 T^{4} - \)\(14\!\cdots\!24\)\( T^{5} + \)\(77\!\cdots\!84\)\( T^{6} - \)\(80\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!18\)\( T^{8} - \)\(80\!\cdots\!40\)\( p^{5} T^{9} + \)\(77\!\cdots\!84\)\( p^{10} T^{10} - \)\(14\!\cdots\!24\)\( p^{15} T^{11} + 1170969803856514292 p^{20} T^{12} - 9428770082944 p^{25} T^{13} + 1555305004 p^{30} T^{14} + 6224 p^{35} T^{15} + p^{40} T^{16} \)
59 \( 1 + 77556 T + 6103413850 T^{2} + 267415908844496 T^{3} + 11785473752724346233 T^{4} + \)\(35\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!14\)\( T^{6} + \)\(26\!\cdots\!64\)\( T^{7} + \)\(81\!\cdots\!04\)\( T^{8} + \)\(26\!\cdots\!64\)\( p^{5} T^{9} + \)\(11\!\cdots\!14\)\( p^{10} T^{10} + \)\(35\!\cdots\!08\)\( p^{15} T^{11} + 11785473752724346233 p^{20} T^{12} + 267415908844496 p^{25} T^{13} + 6103413850 p^{30} T^{14} + 77556 p^{35} T^{15} + p^{40} T^{16} \)
61 \( 1 - 11554 T + 2291818845 T^{2} - 28011632085862 T^{3} + 2733788446130425422 T^{4} - \)\(37\!\cdots\!70\)\( T^{5} + \)\(31\!\cdots\!47\)\( T^{6} - \)\(47\!\cdots\!98\)\( T^{7} + \)\(31\!\cdots\!38\)\( T^{8} - \)\(47\!\cdots\!98\)\( p^{5} T^{9} + \)\(31\!\cdots\!47\)\( p^{10} T^{10} - \)\(37\!\cdots\!70\)\( p^{15} T^{11} + 2733788446130425422 p^{20} T^{12} - 28011632085862 p^{25} T^{13} + 2291818845 p^{30} T^{14} - 11554 p^{35} T^{15} + p^{40} T^{16} \)
67 \( 1 + 20894 T + 2388080605 T^{2} + 98102161763574 T^{3} + 5643738854296309154 T^{4} + \)\(25\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} + \)\(39\!\cdots\!78\)\( T^{7} + \)\(17\!\cdots\!78\)\( T^{8} + \)\(39\!\cdots\!78\)\( p^{5} T^{9} + \)\(10\!\cdots\!03\)\( p^{10} T^{10} + \)\(25\!\cdots\!94\)\( p^{15} T^{11} + 5643738854296309154 p^{20} T^{12} + 98102161763574 p^{25} T^{13} + 2388080605 p^{30} T^{14} + 20894 p^{35} T^{15} + p^{40} T^{16} \)
71 \( 1 + 21648 T + 6518390945 T^{2} + 220930253468240 T^{3} + 25873218849782626409 T^{4} + \)\(92\!\cdots\!36\)\( T^{5} + \)\(73\!\cdots\!54\)\( T^{6} + \)\(23\!\cdots\!68\)\( T^{7} + \)\(15\!\cdots\!66\)\( T^{8} + \)\(23\!\cdots\!68\)\( p^{5} T^{9} + \)\(73\!\cdots\!54\)\( p^{10} T^{10} + \)\(92\!\cdots\!36\)\( p^{15} T^{11} + 25873218849782626409 p^{20} T^{12} + 220930253468240 p^{25} T^{13} + 6518390945 p^{30} T^{14} + 21648 p^{35} T^{15} + p^{40} T^{16} \)
73 \( 1 + 64660 T + 10635628104 T^{2} + 635848241546940 T^{3} + 60609372752114860812 T^{4} + \)\(31\!\cdots\!84\)\( T^{5} + \)\(21\!\cdots\!28\)\( T^{6} + \)\(96\!\cdots\!24\)\( T^{7} + \)\(54\!\cdots\!42\)\( T^{8} + \)\(96\!\cdots\!24\)\( p^{5} T^{9} + \)\(21\!\cdots\!28\)\( p^{10} T^{10} + \)\(31\!\cdots\!84\)\( p^{15} T^{11} + 60609372752114860812 p^{20} T^{12} + 635848241546940 p^{25} T^{13} + 10635628104 p^{30} T^{14} + 64660 p^{35} T^{15} + p^{40} T^{16} \)
79 \( 1 + 22660 T + 10725095318 T^{2} + 278108395968756 T^{3} + 60112235538086853849 T^{4} + \)\(21\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!54\)\( T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(82\!\cdots\!68\)\( T^{8} + \)\(10\!\cdots\!96\)\( p^{5} T^{9} + \)\(24\!\cdots\!54\)\( p^{10} T^{10} + \)\(21\!\cdots\!76\)\( p^{15} T^{11} + 60112235538086853849 p^{20} T^{12} + 278108395968756 p^{25} T^{13} + 10725095318 p^{30} T^{14} + 22660 p^{35} T^{15} + p^{40} T^{16} \)
83 \( 1 + 100390 T + 26540290069 T^{2} + 2307308067513778 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!86\)\( T^{5} + \)\(24\!\cdots\!39\)\( T^{6} + \)\(15\!\cdots\!54\)\( T^{7} + \)\(11\!\cdots\!66\)\( T^{8} + \)\(15\!\cdots\!54\)\( p^{5} T^{9} + \)\(24\!\cdots\!39\)\( p^{10} T^{10} + \)\(24\!\cdots\!86\)\( p^{15} T^{11} + \)\(32\!\cdots\!74\)\( p^{20} T^{12} + 2307308067513778 p^{25} T^{13} + 26540290069 p^{30} T^{14} + 100390 p^{35} T^{15} + p^{40} T^{16} \)
89 \( 1 + 25578 T + 22967459957 T^{2} - 192904433310934 T^{3} + \)\(21\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!58\)\( T^{5} + \)\(11\!\cdots\!95\)\( T^{6} - \)\(15\!\cdots\!30\)\( T^{7} + \)\(51\!\cdots\!30\)\( T^{8} - \)\(15\!\cdots\!30\)\( p^{5} T^{9} + \)\(11\!\cdots\!95\)\( p^{10} T^{10} - \)\(12\!\cdots\!58\)\( p^{15} T^{11} + \)\(21\!\cdots\!98\)\( p^{20} T^{12} - 192904433310934 p^{25} T^{13} + 22967459957 p^{30} T^{14} + 25578 p^{35} T^{15} + p^{40} T^{16} \)
97 \( 1 + 142828 T + 47536469444 T^{2} + 5581613659915992 T^{3} + \)\(10\!\cdots\!62\)\( T^{4} + \)\(10\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!84\)\( p^{5} T^{9} + \)\(13\!\cdots\!76\)\( p^{10} T^{10} + \)\(10\!\cdots\!56\)\( p^{15} T^{11} + \)\(10\!\cdots\!62\)\( p^{20} T^{12} + 5581613659915992 p^{25} T^{13} + 47536469444 p^{30} T^{14} + 142828 p^{35} T^{15} + p^{40} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.31622949854532930839124692631, −3.76211355090268241606048141649, −3.67247399911004841120126356402, −3.59136877154359754826963741551, −3.57635738773756887209810161052, −3.51948244392430793638428746764, −3.29826444171266279514123011585, −3.13336300002015501473016062560, −3.13330553160019967231035485165, −2.84753625680686007572153734585, −2.76196365381324179487364304947, −2.59627031508510415099300795652, −2.54322095085225543478526645127, −2.27769626753586617621426201637, −2.20791522213230542119083461972, −2.17237791311444111259987407087, −2.00400854961734466567857302850, −1.97607257947709297115868743959, −1.65804331352346114337219006402, −1.35405676618669954579554302406, −1.34832759319246584770861638307, −1.26331695637334938689483274012, −1.13401951089283040052713030851, −1.11367988958456383454735177962, −0.994401446011749395080638946479, 0, 0, 0, 0, 0, 0, 0, 0, 0.994401446011749395080638946479, 1.11367988958456383454735177962, 1.13401951089283040052713030851, 1.26331695637334938689483274012, 1.34832759319246584770861638307, 1.35405676618669954579554302406, 1.65804331352346114337219006402, 1.97607257947709297115868743959, 2.00400854961734466567857302850, 2.17237791311444111259987407087, 2.20791522213230542119083461972, 2.27769626753586617621426201637, 2.54322095085225543478526645127, 2.59627031508510415099300795652, 2.76196365381324179487364304947, 2.84753625680686007572153734585, 3.13330553160019967231035485165, 3.13336300002015501473016062560, 3.29826444171266279514123011585, 3.51948244392430793638428746764, 3.57635738773756887209810161052, 3.59136877154359754826963741551, 3.67247399911004841120126356402, 3.76211355090268241606048141649, 4.31622949854532930839124692631

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

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