Properties

Label 16-1050e8-1.1-c2e8-0-10
Degree $16$
Conductor $1.477\times 10^{24}$
Sign $1$
Analytic cond. $4.48946\times 10^{11}$
Root an. cond. $5.34887$
Motivic weight $2$
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  − 12·3-s − 4·4-s + 78·9-s − 4·11-s + 48·12-s + 4·16-s − 24·17-s + 72·19-s + 60·23-s − 360·27-s − 24·29-s + 96·31-s + 48·33-s − 312·36-s − 24·37-s − 112·43-s + 16·44-s + 84·47-s − 48·48-s − 132·49-s + 288·51-s + 44·53-s − 864·57-s + 312·59-s − 204·61-s + 16·64-s + 120·67-s + ⋯
L(s)  = 1  − 4·3-s − 4-s + 26/3·9-s − 0.363·11-s + 4·12-s + 1/4·16-s − 1.41·17-s + 3.78·19-s + 2.60·23-s − 13.3·27-s − 0.827·29-s + 3.09·31-s + 1.45·33-s − 8.66·36-s − 0.648·37-s − 2.60·43-s + 4/11·44-s + 1.78·47-s − 48-s − 2.69·49-s + 5.64·51-s + 0.830·53-s − 15.1·57-s + 5.28·59-s − 3.34·61-s + 1/4·64-s + 1.79·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(4.48946\times 10^{11}\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.425776457\)
\(L(\frac12)\) \(\approx\) \(1.425776457\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
3 \( ( 1 + p T + p T^{2} )^{4} \)
5 \( 1 \)
7 \( 1 + 132 T^{2} + 167 p^{2} T^{4} + 132 p^{4} T^{6} + p^{8} T^{8} \)
good11 \( 1 + 4 T - 426 T^{2} - 952 T^{3} + 112064 T^{4} + 142800 T^{5} - 20280116 T^{6} - 6609908 T^{7} + 2820384747 T^{8} - 6609908 p^{2} T^{9} - 20280116 p^{4} T^{10} + 142800 p^{6} T^{11} + 112064 p^{8} T^{12} - 952 p^{10} T^{13} - 426 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 - 812 T^{2} + 291354 T^{4} - 64773520 T^{6} + 11461503587 T^{8} - 64773520 p^{4} T^{10} + 291354 p^{8} T^{12} - 812 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 + 24 T + 650 T^{2} + 10992 T^{3} + 160192 T^{4} + 2399124 T^{5} + 32567204 T^{6} + 818570952 T^{7} + 12264088027 T^{8} + 818570952 p^{2} T^{9} + 32567204 p^{4} T^{10} + 2399124 p^{6} T^{11} + 160192 p^{8} T^{12} + 10992 p^{10} T^{13} + 650 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 - 72 T + 3028 T^{2} - 93600 T^{3} + 2298945 T^{4} - 49353840 T^{5} + 974880260 T^{6} - 18709788168 T^{7} + 358360390448 T^{8} - 18709788168 p^{2} T^{9} + 974880260 p^{4} T^{10} - 49353840 p^{6} T^{11} + 2298945 p^{8} T^{12} - 93600 p^{10} T^{13} + 3028 p^{12} T^{14} - 72 p^{14} T^{15} + p^{16} T^{16} \)
23 \( 1 - 60 T + 432 T^{2} + 14472 T^{3} + 1091530 T^{4} - 41138292 T^{5} + 91044000 T^{6} - 6476074260 T^{7} + 531075903987 T^{8} - 6476074260 p^{2} T^{9} + 91044000 p^{4} T^{10} - 41138292 p^{6} T^{11} + 1091530 p^{8} T^{12} + 14472 p^{10} T^{13} + 432 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16} \)
29 \( ( 1 + 12 T + 2040 T^{2} + 5892 T^{3} + 1907222 T^{4} + 5892 p^{2} T^{5} + 2040 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 96 T + 7300 T^{2} - 405888 T^{3} + 20288250 T^{4} - 28540512 p T^{5} + 34750935632 T^{6} - 39793390752 p T^{7} + 39789694494419 T^{8} - 39793390752 p^{3} T^{9} + 34750935632 p^{4} T^{10} - 28540512 p^{7} T^{11} + 20288250 p^{8} T^{12} - 405888 p^{10} T^{13} + 7300 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} \)
37 \( 1 + 24 T - 3150 T^{2} - 62256 T^{3} + 4743841 T^{4} + 35089200 T^{5} - 10044719118 T^{6} + 10547453160 T^{7} + 19118891659236 T^{8} + 10547453160 p^{2} T^{9} - 10044719118 p^{4} T^{10} + 35089200 p^{6} T^{11} + 4743841 p^{8} T^{12} - 62256 p^{10} T^{13} - 3150 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \)
41 \( 1 - 8684 T^{2} + 37350972 T^{4} - 104578299436 T^{6} + 123169986458 p^{2} T^{8} - 104578299436 p^{4} T^{10} + 37350972 p^{8} T^{12} - 8684 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 56 T + 2892 T^{2} - 57080 T^{3} - 2590714 T^{4} - 57080 p^{2} T^{5} + 2892 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 84 T + 7246 T^{2} - 411096 T^{3} + 21782184 T^{4} - 820450080 T^{5} + 16949559308 T^{6} - 365731926996 T^{7} - 7613654937685 T^{8} - 365731926996 p^{2} T^{9} + 16949559308 p^{4} T^{10} - 820450080 p^{6} T^{11} + 21782184 p^{8} T^{12} - 411096 p^{10} T^{13} + 7246 p^{12} T^{14} - 84 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 - 44 T - 6194 T^{2} + 119192 T^{3} + 25314944 T^{4} + 57040312 T^{5} - 90568544740 T^{6} - 129300820708 T^{7} + 259778649246667 T^{8} - 129300820708 p^{2} T^{9} - 90568544740 p^{4} T^{10} + 57040312 p^{6} T^{11} + 25314944 p^{8} T^{12} + 119192 p^{10} T^{13} - 6194 p^{12} T^{14} - 44 p^{14} T^{15} + p^{16} T^{16} \)
59 \( 1 - 312 T + 52130 T^{2} - 6140784 T^{3} + 573933472 T^{4} - 46040844252 T^{5} + 3331577620436 T^{6} - 222192183834504 T^{7} + 13673835884421787 T^{8} - 222192183834504 p^{2} T^{9} + 3331577620436 p^{4} T^{10} - 46040844252 p^{6} T^{11} + 573933472 p^{8} T^{12} - 6140784 p^{10} T^{13} + 52130 p^{12} T^{14} - 312 p^{14} T^{15} + p^{16} T^{16} \)
61 \( 1 + 204 T + 29950 T^{2} + 3279912 T^{3} + 303356793 T^{4} + 24117206328 T^{5} + 1743285323678 T^{6} + 115555686856116 T^{7} + 7243930491862772 T^{8} + 115555686856116 p^{2} T^{9} + 1743285323678 p^{4} T^{10} + 24117206328 p^{6} T^{11} + 303356793 p^{8} T^{12} + 3279912 p^{10} T^{13} + 29950 p^{12} T^{14} + 204 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 - 120 T - 7516 T^{2} + 645360 T^{3} + 127684489 T^{4} - 5775540120 T^{5} - 738336863500 T^{6} + 2722631928240 T^{7} + 4851695923876672 T^{8} + 2722631928240 p^{2} T^{9} - 738336863500 p^{4} T^{10} - 5775540120 p^{6} T^{11} + 127684489 p^{8} T^{12} + 645360 p^{10} T^{13} - 7516 p^{12} T^{14} - 120 p^{14} T^{15} + p^{16} T^{16} \)
71 \( ( 1 + 32 T + 8676 T^{2} + 266080 T^{3} + 63690374 T^{4} + 266080 p^{2} T^{5} + 8676 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 84 T + 7974 T^{2} - 472248 T^{3} + 12876001 T^{4} - 302804952 T^{5} + 150489687510 T^{6} - 23139945244956 T^{7} + 1753519084103172 T^{8} - 23139945244956 p^{2} T^{9} + 150489687510 p^{4} T^{10} - 302804952 p^{6} T^{11} + 12876001 p^{8} T^{12} - 472248 p^{10} T^{13} + 7974 p^{12} T^{14} - 84 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 + 144 T - 612 T^{2} - 13824 p T^{3} - 51094919 T^{4} - 2106377280 T^{5} - 285913987380 T^{6} + 30587071339440 T^{7} + 6524654383140480 T^{8} + 30587071339440 p^{2} T^{9} - 285913987380 p^{4} T^{10} - 2106377280 p^{6} T^{11} - 51094919 p^{8} T^{12} - 13824 p^{11} T^{13} - 612 p^{12} T^{14} + 144 p^{14} T^{15} + p^{16} T^{16} \)
83 \( 1 - 44180 T^{2} + 901887852 T^{4} - 11200337241460 T^{6} + 93097977064556858 T^{8} - 11200337241460 p^{4} T^{10} + 901887852 p^{8} T^{12} - 44180 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 + 336 T + 78946 T^{2} + 13881504 T^{3} + 2085368016 T^{4} + 269643799980 T^{5} + 31112060435732 T^{6} + 3208326913520640 T^{7} + 300257635897974779 T^{8} + 3208326913520640 p^{2} T^{9} + 31112060435732 p^{4} T^{10} + 269643799980 p^{6} T^{11} + 2085368016 p^{8} T^{12} + 13881504 p^{10} T^{13} + 78946 p^{12} T^{14} + 336 p^{14} T^{15} + p^{16} T^{16} \)
97 \( 1 - 46260 T^{2} + 993971866 T^{4} - 13674806149680 T^{6} + 142735721327901795 T^{8} - 13674806149680 p^{4} T^{10} + 993971866 p^{8} T^{12} - 46260 p^{12} T^{14} + p^{16} T^{16} \)
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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

−4.14046358117295050867603799079, −4.07967876094829221352878970957, −3.85609298142525543882781915298, −3.79410702086716083869043106178, −3.59172109975424691215827103543, −3.28209484416867968944455627230, −3.23742037180951737946421089918, −2.99881043454060676834182279614, −2.96151759463733775356761531605, −2.87911250966596905999210182094, −2.81737135814270523045139638752, −2.49657470704118397418356840207, −2.33041053714926577434407029990, −2.24831264125018986694286803552, −1.75839932381316687749551591279, −1.63268189668432856803070207502, −1.50450669330328361194107488745, −1.35698630428371312464232035447, −1.28109740176953547811055413450, −1.11497410341314863390173436378, −0.77786244306994645498414238925, −0.67881349049474286780492336722, −0.43686286437836542698765397393, −0.41262566532617654963138977256, −0.24405835152442709919695702893, 0.24405835152442709919695702893, 0.41262566532617654963138977256, 0.43686286437836542698765397393, 0.67881349049474286780492336722, 0.77786244306994645498414238925, 1.11497410341314863390173436378, 1.28109740176953547811055413450, 1.35698630428371312464232035447, 1.50450669330328361194107488745, 1.63268189668432856803070207502, 1.75839932381316687749551591279, 2.24831264125018986694286803552, 2.33041053714926577434407029990, 2.49657470704118397418356840207, 2.81737135814270523045139638752, 2.87911250966596905999210182094, 2.96151759463733775356761531605, 2.99881043454060676834182279614, 3.23742037180951737946421089918, 3.28209484416867968944455627230, 3.59172109975424691215827103543, 3.79410702086716083869043106178, 3.85609298142525543882781915298, 4.07967876094829221352878970957, 4.14046358117295050867603799079

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

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