Properties

Label 16-11e16-1.1-c3e8-0-0
Degree $16$
Conductor $4.595\times 10^{16}$
Sign $1$
Analytic cond. $6.74856\times 10^{6}$
Root an. cond. $2.67193$
Motivic weight $3$
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  − 2·2-s + 2·3-s + 14·4-s − 2·5-s − 4·6-s − 20·7-s − 16·8-s + 7·9-s + 4·10-s + 28·12-s − 80·13-s + 40·14-s − 4·15-s + 64·16-s + 124·17-s − 14·18-s − 72·19-s − 28·20-s − 40·21-s − 392·23-s − 32·24-s + 59·25-s + 160·26-s + 54·27-s − 280·28-s − 144·29-s + 8·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.384·3-s + 7/4·4-s − 0.178·5-s − 0.272·6-s − 1.07·7-s − 0.707·8-s + 7/27·9-s + 0.126·10-s + 0.673·12-s − 1.70·13-s + 0.763·14-s − 0.0688·15-s + 16-s + 1.76·17-s − 0.183·18-s − 0.869·19-s − 0.313·20-s − 0.415·21-s − 3.55·23-s − 0.272·24-s + 0.471·25-s + 1.20·26-s + 0.384·27-s − 1.88·28-s − 0.922·29-s + 0.0486·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(11^{16}\)
Sign: $1$
Analytic conductor: \(6.74856\times 10^{6}\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 11^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.4933803557\)
\(L(\frac12)\) \(\approx\) \(0.4933803557\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + p T - 5 p T^{2} - p^{5} T^{3} + 11 p^{2} T^{4} + p^{8} T^{5} + 3 p^{3} T^{6} - 49 p^{4} T^{7} - 39 p^{4} T^{8} - 49 p^{7} T^{9} + 3 p^{9} T^{10} + p^{17} T^{11} + 11 p^{14} T^{12} - p^{20} T^{13} - 5 p^{19} T^{14} + p^{22} T^{15} + p^{24} T^{16} \)
3 \( 1 - 2 T - p T^{2} - 34 T^{3} - 532 T^{4} - 4936 T^{5} + 5873 p T^{6} + 52828 T^{7} + 425383 T^{8} + 52828 p^{3} T^{9} + 5873 p^{7} T^{10} - 4936 p^{9} T^{11} - 532 p^{12} T^{12} - 34 p^{15} T^{13} - p^{19} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} \)
5 \( 1 + 2 T - 11 p T^{2} + 22 T^{3} - 11836 T^{4} + 122548 T^{5} + 361659 p T^{6} - 6916492 T^{7} + 95052111 T^{8} - 6916492 p^{3} T^{9} + 361659 p^{7} T^{10} + 122548 p^{9} T^{11} - 11836 p^{12} T^{12} + 22 p^{15} T^{13} - 11 p^{19} T^{14} + 2 p^{21} T^{15} + p^{24} T^{16} \)
7 \( 1 + 20 T - 338 T^{2} - 14660 T^{3} - 24205 T^{4} + 877220 T^{5} - 25394548 T^{6} + 403408720 T^{7} + 35674774069 T^{8} + 403408720 p^{3} T^{9} - 25394548 p^{6} T^{10} + 877220 p^{9} T^{11} - 24205 p^{12} T^{12} - 14660 p^{15} T^{13} - 338 p^{18} T^{14} + 20 p^{21} T^{15} + p^{24} T^{16} \)
13 \( 1 + 80 T + 1606 T^{2} - 79280 T^{3} - 4807573 T^{4} + 213824720 T^{5} + 18467374508 T^{6} - 9995360000 T^{7} - 28545152779595 T^{8} - 9995360000 p^{3} T^{9} + 18467374508 p^{6} T^{10} + 213824720 p^{9} T^{11} - 4807573 p^{12} T^{12} - 79280 p^{15} T^{13} + 1606 p^{18} T^{14} + 80 p^{21} T^{15} + p^{24} T^{16} \)
17 \( 1 - 124 T + 2138 T^{2} + 767188 T^{3} - 72029437 T^{4} + 3781849588 T^{5} - 34409920284 T^{6} - 20434150643056 T^{7} + 2424020558872533 T^{8} - 20434150643056 p^{3} T^{9} - 34409920284 p^{6} T^{10} + 3781849588 p^{9} T^{11} - 72029437 p^{12} T^{12} + 767188 p^{15} T^{13} + 2138 p^{18} T^{14} - 124 p^{21} T^{15} + p^{24} T^{16} \)
19 \( 1 + 72 T + 970 T^{2} + 260280 T^{3} + 3163755 T^{4} + 10300665096 T^{5} + 811220076212 T^{6} + 4318154945280 T^{7} + 1830546969989045 T^{8} + 4318154945280 p^{3} T^{9} + 811220076212 p^{6} T^{10} + 10300665096 p^{9} T^{11} + 3163755 p^{12} T^{12} + 260280 p^{15} T^{13} + 970 p^{18} T^{14} + 72 p^{21} T^{15} + p^{24} T^{16} \)
23 \( ( 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
29 \( 1 + 144 T - 23818 T^{2} - 6333552 T^{3} + 60062667 T^{4} + 142076993040 T^{5} + 9707020830988 T^{6} - 954001866369024 T^{7} - 106612877125667275 T^{8} - 954001866369024 p^{3} T^{9} + 9707020830988 p^{6} T^{10} + 142076993040 p^{9} T^{11} + 60062667 p^{12} T^{12} - 6333552 p^{15} T^{13} - 23818 p^{18} T^{14} + 144 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 34 T - 56363 T^{2} + 2859094 T^{3} + 2291668916 T^{4} - 40252636720 T^{5} - 83319503916757 T^{6} + 289480400339500 T^{7} + 2789519267810990767 T^{8} + 289480400339500 p^{3} T^{9} - 83319503916757 p^{6} T^{10} - 40252636720 p^{9} T^{11} + 2291668916 p^{12} T^{12} + 2859094 p^{15} T^{13} - 56363 p^{18} T^{14} - 34 p^{21} T^{15} + p^{24} T^{16} \)
37 \( 1 + 54 T - 98927 T^{2} - 8106318 T^{3} + 7219259028 T^{4} + 131258461932 T^{5} - 496496327366449 T^{6} + 366601843468476 T^{7} + 32420854709078338943 T^{8} + 366601843468476 p^{3} T^{9} - 496496327366449 p^{6} T^{10} + 131258461932 p^{9} T^{11} + 7219259028 p^{12} T^{12} - 8106318 p^{15} T^{13} - 98927 p^{18} T^{14} + 54 p^{21} T^{15} + p^{24} T^{16} \)
41 \( 1 + 536 T + 77678 T^{2} - 33778184 T^{3} - 19337190253 T^{4} - 5181455663720 T^{5} - 340637740165668 T^{6} + 349646609327660288 T^{7} + \)\(15\!\cdots\!45\)\( T^{8} + 349646609327660288 p^{3} T^{9} - 340637740165668 p^{6} T^{10} - 5181455663720 p^{9} T^{11} - 19337190253 p^{12} T^{12} - 33778184 p^{15} T^{13} + 77678 p^{18} T^{14} + 536 p^{21} T^{15} + p^{24} T^{16} \)
43 \( ( 1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
47 \( 1 - 272 T - 108958 T^{2} + 51156944 T^{3} + 2920331171 T^{4} - 5591002519312 T^{5} + 716318472024900 T^{6} + 194089111806257152 T^{7} - 57426744803672822331 T^{8} + 194089111806257152 p^{3} T^{9} + 716318472024900 p^{6} T^{10} - 5591002519312 p^{9} T^{11} + 2920331171 p^{12} T^{12} + 51156944 p^{15} T^{13} - 108958 p^{18} T^{14} - 272 p^{21} T^{15} + p^{24} T^{16} \)
53 \( 1 - 492 T - 106798 T^{2} + 150937236 T^{3} - 23129955237 T^{4} + 4296299406564 T^{5} - 2733280203198476 T^{6} - 1805258123973305712 T^{7} + \)\(20\!\cdots\!13\)\( T^{8} - 1805258123973305712 p^{3} T^{9} - 2733280203198476 p^{6} T^{10} + 4296299406564 p^{9} T^{11} - 23129955237 p^{12} T^{12} + 150937236 p^{15} T^{13} - 106798 p^{18} T^{14} - 492 p^{21} T^{15} + p^{24} T^{16} \)
59 \( 1 + 634 T - 57019 T^{2} - 196929910 T^{3} - 58310479732 T^{4} + 4690390137800 T^{5} + 6499551726981819 T^{6} + 2681928432382897204 T^{7} + \)\(17\!\cdots\!23\)\( T^{8} + 2681928432382897204 p^{3} T^{9} + 6499551726981819 p^{6} T^{10} + 4690390137800 p^{9} T^{11} - 58310479732 p^{12} T^{12} - 196929910 p^{15} T^{13} - 57019 p^{18} T^{14} + 634 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 + 840 T + 176806 T^{2} - 105005880 T^{3} - 73061471925 T^{4} + 6883771568520 T^{5} + 15286879488839084 T^{6} + 680628509960031360 T^{7} - \)\(24\!\cdots\!71\)\( T^{8} + 680628509960031360 p^{3} T^{9} + 15286879488839084 p^{6} T^{10} + 6883771568520 p^{9} T^{11} - 73061471925 p^{12} T^{12} - 105005880 p^{15} T^{13} + 176806 p^{18} T^{14} + 840 p^{21} T^{15} + p^{24} T^{16} \)
67 \( ( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
71 \( 1 - 678 T - 353731 T^{2} + 548661330 T^{3} - 47836604172 T^{4} + 14058061430640 T^{5} - 58447478076692429 T^{6} - 30483114697404462492 T^{7} + \)\(70\!\cdots\!03\)\( T^{8} - 30483114697404462492 p^{3} T^{9} - 58447478076692429 p^{6} T^{10} + 14058061430640 p^{9} T^{11} - 47836604172 p^{12} T^{12} + 548661330 p^{15} T^{13} - 353731 p^{18} T^{14} - 678 p^{21} T^{15} + p^{24} T^{16} \)
73 \( 1 - 400 T - 962 T^{2} - 90837200 T^{3} - 52601780845 T^{4} - 196521796416400 T^{5} + 101356075953582908 T^{6} + 13022083185194310400 T^{7} + \)\(17\!\cdots\!09\)\( T^{8} + 13022083185194310400 p^{3} T^{9} + 101356075953582908 p^{6} T^{10} - 196521796416400 p^{9} T^{11} - 52601780845 p^{12} T^{12} - 90837200 p^{15} T^{13} - 962 p^{18} T^{14} - 400 p^{21} T^{15} + p^{24} T^{16} \)
79 \( 1 + 4 p T + 379822 T^{2} + 4611308 p T^{3} + 27599385827 T^{4} + 127903538260 p T^{5} - 24653070792693172 T^{6} - 1129324475166871984 p T^{7} - \)\(40\!\cdots\!35\)\( T^{8} - 1129324475166871984 p^{4} T^{9} - 24653070792693172 p^{6} T^{10} + 127903538260 p^{10} T^{11} + 27599385827 p^{12} T^{12} + 4611308 p^{16} T^{13} + 379822 p^{18} T^{14} + 4 p^{22} T^{15} + p^{24} T^{16} \)
83 \( 1 + 468 T - 936106 T^{2} - 711102132 T^{3} + 546823028523 T^{4} + 138605912443908 T^{5} - 451021583584008644 T^{6} + 7629875717642150544 T^{7} + \)\(38\!\cdots\!53\)\( T^{8} + 7629875717642150544 p^{3} T^{9} - 451021583584008644 p^{6} T^{10} + 138605912443908 p^{9} T^{11} + 546823028523 p^{12} T^{12} - 711102132 p^{15} T^{13} - 936106 p^{18} T^{14} + 468 p^{21} T^{15} + p^{24} T^{16} \)
89 \( ( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
97 \( 1 + 2194 T + 1847089 T^{2} - 453686290 T^{3} - 2914560447844 T^{4} - 3260498846137988 T^{5} - 954390839981627665 T^{6} + \)\(21\!\cdots\!08\)\( T^{7} + \)\(33\!\cdots\!27\)\( T^{8} + \)\(21\!\cdots\!08\)\( p^{3} T^{9} - 954390839981627665 p^{6} T^{10} - 3260498846137988 p^{9} T^{11} - 2914560447844 p^{12} T^{12} - 453686290 p^{15} T^{13} + 1847089 p^{18} T^{14} + 2194 p^{21} T^{15} + p^{24} T^{16} \)
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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

−5.82200617313300127203682278126, −5.68639523871673855980070030525, −5.62511967998897306588570240233, −5.10524557464090069440185987082, −5.07983402338143609667550359041, −4.83360414669785642417351681811, −4.80941548048462734144191655125, −4.38126842857172664856605856413, −4.14578433025132730272360736925, −3.91440242997707404642592776207, −3.88892692291587505519364114149, −3.79922818328960131024151482187, −3.42118731669303917393884140558, −3.28365430240730959046458589908, −2.97764375988463407796548786000, −2.72493225577288563438372508662, −2.54100542514602026683780887478, −2.36614030496348924347731834874, −2.11045685231451057465266713155, −1.90292313021274514585857725482, −1.74658442665555663915037583988, −1.20948828150126560026565782908, −1.12867717999534965530719790410, −0.41413139738033480955694756895, −0.12597261921877155317781230356, 0.12597261921877155317781230356, 0.41413139738033480955694756895, 1.12867717999534965530719790410, 1.20948828150126560026565782908, 1.74658442665555663915037583988, 1.90292313021274514585857725482, 2.11045685231451057465266713155, 2.36614030496348924347731834874, 2.54100542514602026683780887478, 2.72493225577288563438372508662, 2.97764375988463407796548786000, 3.28365430240730959046458589908, 3.42118731669303917393884140558, 3.79922818328960131024151482187, 3.88892692291587505519364114149, 3.91440242997707404642592776207, 4.14578433025132730272360736925, 4.38126842857172664856605856413, 4.80941548048462734144191655125, 4.83360414669785642417351681811, 5.07983402338143609667550359041, 5.10524557464090069440185987082, 5.62511967998897306588570240233, 5.68639523871673855980070030525, 5.82200617313300127203682278126

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

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