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Properties

Label 16-847e8-1.1-c1e8-0-10
Degree $16$
Conductor $2.649\times 10^{23}$
Sign $1$
Analytic cond. $4.37808\times 10^{6}$
Root an. cond. $2.60064$
Motivic weight $1$
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 + 3-s + 4-s + 6-s − 2·7-s + 2·8-s + 3·9-s + 12-s + 6·13-s − 2·14-s + 4·16-s + 9·17-s + 3·18-s − 6·19-s − 2·21-s − 36·23-s + 2·24-s − 3·25-s + 6·26-s + 3·27-s − 2·28-s + 13·29-s − 2·31-s + 19·32-s + 9·34-s + 3·36-s − 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.707·8-s + 9-s + 0.288·12-s + 1.66·13-s − 0.534·14-s + 16-s + 2.18·17-s + 0.707·18-s − 1.37·19-s − 0.436·21-s − 7.50·23-s + 0.408·24-s − 3/5·25-s + 1.17·26-s + 0.577·27-s − 0.377·28-s + 2.41·29-s − 0.359·31-s + 3.35·32-s + 1.54·34-s + 1/2·36-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4.37808\times 10^{6}\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.12426825\)
\(L(\frac12)\) \(\approx\) \(14.12426825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( 1 \)
good2 \( 1 - T - T^{3} - T^{4} - 13 T^{5} + p^{4} T^{6} + 3 T^{7} + 11 T^{8} + 3 p T^{9} + p^{6} T^{10} - 13 p^{3} T^{11} - p^{4} T^{12} - p^{5} T^{13} - p^{7} T^{15} + p^{8} T^{16} \)
3 \( 1 - T - 2 T^{2} + 2 T^{3} - 2 T^{4} - 11 p T^{5} + 17 p T^{6} + 4 p^{2} T^{7} - 8 p^{2} T^{8} + 4 p^{3} T^{9} + 17 p^{3} T^{10} - 11 p^{4} T^{11} - 2 p^{4} T^{12} + 2 p^{5} T^{13} - 2 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
5 \( 1 + 3 T^{2} - 16 T^{4} - 123 T^{6} + 31 T^{8} - 123 p^{2} T^{10} - 16 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 6 T + 14 T^{2} - 30 T^{3} + 171 T^{4} - 1518 T^{5} + 5320 T^{6} - 6792 T^{7} + 13217 T^{8} - 6792 p T^{9} + 5320 p^{2} T^{10} - 1518 p^{3} T^{11} + 171 p^{4} T^{12} - 30 p^{5} T^{13} + 14 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 9 T + 30 T^{2} + 36 T^{3} - 766 T^{4} + 189 p T^{5} - 237 p T^{6} - 72 p^{2} T^{7} + 424 p^{2} T^{8} - 72 p^{3} T^{9} - 237 p^{3} T^{10} + 189 p^{4} T^{11} - 766 p^{4} T^{12} + 36 p^{5} T^{13} + 30 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
19 \( ( 1 + 3 T - 10 T^{2} - 87 T^{3} - 71 T^{4} - 87 p T^{5} - 10 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 9 T + 63 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \)
29 \( 1 - 13 T + 72 T^{2} - 52 T^{3} - 2248 T^{4} + 16445 T^{5} - 36677 T^{6} - 227136 T^{7} + 2201240 T^{8} - 227136 p T^{9} - 36677 p^{2} T^{10} + 16445 p^{3} T^{11} - 2248 p^{4} T^{12} - 52 p^{5} T^{13} + 72 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 + T - 30 T^{2} - 61 T^{3} + 869 T^{4} - 61 p T^{5} - 30 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 4 T - 10 T^{2} + 4 T^{3} - 501 T^{4} + 19492 T^{5} + 105276 T^{6} - 165312 T^{7} + 172261 T^{8} - 165312 p T^{9} + 105276 p^{2} T^{10} + 19492 p^{3} T^{11} - 501 p^{4} T^{12} + 4 p^{5} T^{13} - 10 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( ( 1 - 7 T + 8 T^{2} + 231 T^{3} - 1945 T^{4} + 231 p T^{5} + 8 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( 1 + 7 T - 28 T^{2} - 406 T^{3} - 592 T^{4} + 33551 T^{5} + 208719 T^{6} - 420308 T^{7} - 6667512 T^{8} - 420308 p T^{9} + 208719 p^{2} T^{10} + 33551 p^{3} T^{11} - 592 p^{4} T^{12} - 406 p^{5} T^{13} - 28 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 15 T + 92 T^{2} - 180 T^{3} - 420 T^{4} - 4065 T^{5} + 1507 T^{6} + 1357560 T^{7} - 16152376 T^{8} + 1357560 p T^{9} + 1507 p^{2} T^{10} - 4065 p^{3} T^{11} - 420 p^{4} T^{12} - 180 p^{5} T^{13} + 92 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 17 T + 102 T^{2} - 442 T^{3} - 13018 T^{4} - 109735 T^{5} - 229517 T^{6} + 5100204 T^{7} + 64874600 T^{8} + 5100204 p T^{9} - 229517 p^{2} T^{10} - 109735 p^{3} T^{11} - 13018 p^{4} T^{12} - 442 p^{5} T^{13} + 102 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 5 T - 100 T^{2} - 820 T^{3} + 6204 T^{4} + 14315 T^{5} - 581925 T^{6} + 244440 T^{7} + 55243816 T^{8} + 244440 p T^{9} - 581925 p^{2} T^{10} + 14315 p^{3} T^{11} + 6204 p^{4} T^{12} - 820 p^{5} T^{13} - 100 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 + T + 53 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
71 \( 1 - 5 T - 120 T^{2} + 970 T^{3} + 9284 T^{4} - 20165 T^{5} - 984785 T^{6} - 337740 T^{7} + 104840456 T^{8} - 337740 p T^{9} - 984785 p^{2} T^{10} - 20165 p^{3} T^{11} + 9284 p^{4} T^{12} + 970 p^{5} T^{13} - 120 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
73 \( ( 1 + 5 T - 48 T^{2} - 605 T^{3} + 479 T^{4} - 605 p T^{5} - 48 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 13 T - 28 T^{2} - 1898 T^{3} - 12048 T^{4} - 72995 T^{5} - 349977 T^{6} + 9302436 T^{7} + 190102840 T^{8} + 9302436 p T^{9} - 349977 p^{2} T^{10} - 72995 p^{3} T^{11} - 12048 p^{4} T^{12} - 1898 p^{5} T^{13} - 28 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 - 3 T - 74 T^{2} + 471 T^{3} + 4729 T^{4} + 471 p T^{5} - 74 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 3 T - 83 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( 1 - 181 T^{2} + 23352 T^{4} - 2523683 T^{6} + 237067655 T^{8} - 2523683 p^{2} T^{10} + 23352 p^{4} T^{12} - 181 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.45510331621139530009852795959, −4.20745020378437958043903235560, −4.02556688116527420809139997008, −3.99759095526103114674106524841, −3.94073960660044457371291595438, −3.90781908218206452195894381778, −3.46789675293851133003824933939, −3.34576048455100484320902694988, −3.32227743727258116872261964191, −3.28588373073210315149998617848, −3.02879663177000195087014911692, −2.97593344118493940666863206729, −2.96433009750334683953563148762, −2.63616850198049821790930752401, −2.15004471328163493229844300439, −2.04230052972315135588477729020, −2.00695346707586093493572999744, −1.91085040168415555166817207971, −1.84082266078244048876466171961, −1.66675116203135327213068283273, −1.64989965660975550792934066955, −0.936031769813416258126229142235, −0.816540821389880715799656404903, −0.67400751970587116310113809016, −0.39316857343779304136516581320, 0.39316857343779304136516581320, 0.67400751970587116310113809016, 0.816540821389880715799656404903, 0.936031769813416258126229142235, 1.64989965660975550792934066955, 1.66675116203135327213068283273, 1.84082266078244048876466171961, 1.91085040168415555166817207971, 2.00695346707586093493572999744, 2.04230052972315135588477729020, 2.15004471328163493229844300439, 2.63616850198049821790930752401, 2.96433009750334683953563148762, 2.97593344118493940666863206729, 3.02879663177000195087014911692, 3.28588373073210315149998617848, 3.32227743727258116872261964191, 3.34576048455100484320902694988, 3.46789675293851133003824933939, 3.90781908218206452195894381778, 3.94073960660044457371291595438, 3.99759095526103114674106524841, 4.02556688116527420809139997008, 4.20745020378437958043903235560, 4.45510331621139530009852795959

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

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