Properties

Label 16-1050e8-1.1-c2e8-0-4
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  − 8·2-s + 32·4-s − 80·8-s − 32·11-s − 40·13-s + 120·16-s + 40·17-s + 256·22-s + 8·23-s + 320·26-s + 96·31-s − 32·32-s − 320·34-s − 112·37-s − 160·41-s + 64·43-s − 1.02e3·44-s − 64·46-s − 64·47-s − 1.28e3·52-s − 80·53-s − 128·61-s − 768·62-s − 384·64-s + 304·67-s + 1.28e3·68-s + 240·71-s + ⋯
L(s)  = 1  − 4·2-s + 8·4-s − 10·8-s − 2.90·11-s − 3.07·13-s + 15/2·16-s + 2.35·17-s + 11.6·22-s + 8/23·23-s + 12.3·26-s + 3.09·31-s − 32-s − 9.41·34-s − 3.02·37-s − 3.90·41-s + 1.48·43-s − 23.2·44-s − 1.39·46-s − 1.36·47-s − 24.6·52-s − 1.50·53-s − 2.09·61-s − 12.3·62-s − 6·64-s + 4.53·67-s + 18.8·68-s + 3.38·71-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

\(L(\frac{3}{2})\) \(\approx\) \(0.03800806584\)
\(L(\frac12)\) \(\approx\) \(0.03800806584\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T + p T^{2} )^{4} \)
3 \( ( 1 + p^{2} T^{4} )^{2} \)
5 \( 1 \)
7 \( ( 1 + p^{2} T^{4} )^{2} \)
good11 \( ( 1 + 16 T + 402 T^{2} + 5648 T^{3} + 68627 T^{4} + 5648 p^{2} T^{5} + 402 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( 1 + 40 T + 800 T^{2} + 13480 T^{3} + 226828 T^{4} + 274360 p T^{5} + 52060000 T^{6} + 766981080 T^{7} + 10735936038 T^{8} + 766981080 p^{2} T^{9} + 52060000 p^{4} T^{10} + 274360 p^{7} T^{11} + 226828 p^{8} T^{12} + 13480 p^{10} T^{13} + 800 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} \)
17 \( 1 - 40 T + 800 T^{2} + 1240 T^{3} - 458548 T^{4} + 10793320 T^{5} - 64125600 T^{6} - 2517543960 T^{7} + 75225125734 T^{8} - 2517543960 p^{2} T^{9} - 64125600 p^{4} T^{10} + 10793320 p^{6} T^{11} - 458548 p^{8} T^{12} + 1240 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} \)
19 \( 1 + 72 T^{2} + 304276 T^{4} - 23053608 T^{6} + 42272435814 T^{8} - 23053608 p^{4} T^{10} + 304276 p^{8} T^{12} + 72 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 8 T + 32 T^{2} - 5072 T^{3} + 314494 T^{4} + 536056 T^{5} - 2816 p^{2} T^{6} - 6265992 p T^{7} + 74059675971 T^{8} - 6265992 p^{3} T^{9} - 2816 p^{6} T^{10} + 536056 p^{6} T^{11} + 314494 p^{8} T^{12} - 5072 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \)
29 \( 1 - 1660 T^{2} + 2334490 T^{4} - 1765266064 T^{6} + 1712096613955 T^{8} - 1765266064 p^{4} T^{10} + 2334490 p^{8} T^{12} - 1660 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 48 T + 2652 T^{2} - 71760 T^{3} + 2870342 T^{4} - 71760 p^{2} T^{5} + 2652 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( 1 + 112 T + 6272 T^{2} + 306208 T^{3} + 12269950 T^{4} + 348054448 T^{5} + 8906641408 T^{6} + 212902064592 T^{7} + 5170693212483 T^{8} + 212902064592 p^{2} T^{9} + 8906641408 p^{4} T^{10} + 348054448 p^{6} T^{11} + 12269950 p^{8} T^{12} + 306208 p^{10} T^{13} + 6272 p^{12} T^{14} + 112 p^{14} T^{15} + p^{16} T^{16} \)
41 \( ( 1 + 80 T + 7812 T^{2} + 382960 T^{3} + 20543558 T^{4} + 382960 p^{2} T^{5} + 7812 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 64 T + 2048 T^{2} - 131104 T^{3} + 10999246 T^{4} - 418295968 T^{5} + 12838615552 T^{6} - 810563773728 T^{7} + 51057140477331 T^{8} - 810563773728 p^{2} T^{9} + 12838615552 p^{4} T^{10} - 418295968 p^{6} T^{11} + 10999246 p^{8} T^{12} - 131104 p^{10} T^{13} + 2048 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} \)
47 \( 1 + 64 T + 2048 T^{2} + 185392 T^{3} + 5922124 T^{4} + 278314096 T^{5} + 22868689024 T^{6} + 896687580960 T^{7} + 41364773451366 T^{8} + 896687580960 p^{2} T^{9} + 22868689024 p^{4} T^{10} + 278314096 p^{6} T^{11} + 5922124 p^{8} T^{12} + 185392 p^{10} T^{13} + 2048 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 + 80 T + 3200 T^{2} + 280016 T^{3} + 30334276 T^{4} + 1292348912 T^{5} + 45522709888 T^{6} + 3555004769904 T^{7} + 277914756020550 T^{8} + 3555004769904 p^{2} T^{9} + 45522709888 p^{4} T^{10} + 1292348912 p^{6} T^{11} + 30334276 p^{8} T^{12} + 280016 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \)
59 \( 1 - 5784 T^{2} + 22133140 T^{4} - 105230012616 T^{6} + 515734904030694 T^{8} - 105230012616 p^{4} T^{10} + 22133140 p^{8} T^{12} - 5784 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 64 T + 3492 T^{2} - 371008 T^{3} - 23720218 T^{4} - 371008 p^{2} T^{5} + 3492 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 304 T + 46208 T^{2} - 5672320 T^{3} + 648244222 T^{4} - 62771615440 T^{5} + 5216109174784 T^{6} - 407407394859312 T^{7} + 29229372215112675 T^{8} - 407407394859312 p^{2} T^{9} + 5216109174784 p^{4} T^{10} - 62771615440 p^{6} T^{11} + 648244222 p^{8} T^{12} - 5672320 p^{10} T^{13} + 46208 p^{12} T^{14} - 304 p^{14} T^{15} + p^{16} T^{16} \)
71 \( ( 1 - 120 T + 15402 T^{2} - 1319424 T^{3} + 121585547 T^{4} - 1319424 p^{2} T^{5} + 15402 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 + 24 T + 288 T^{2} + 322296 T^{3} - 21923636 T^{4} - 1557817848 T^{5} + 20863734624 T^{6} - 2591055821592 T^{7} - 306107011026714 T^{8} - 2591055821592 p^{2} T^{9} + 20863734624 p^{4} T^{10} - 1557817848 p^{6} T^{11} - 21923636 p^{8} T^{12} + 322296 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} \)
79 \( 1 - 22460 T^{2} + 303809850 T^{4} - 2910480225424 T^{6} + 20547055547350115 T^{8} - 2910480225424 p^{4} T^{10} + 303809850 p^{8} T^{12} - 22460 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 + 64 T + 2048 T^{2} + 473152 T^{3} + 190128196 T^{4} + 8448005824 T^{5} + 263226234880 T^{6} + 59283186442944 T^{7} + 13346227966213830 T^{8} + 59283186442944 p^{2} T^{9} + 263226234880 p^{4} T^{10} + 8448005824 p^{6} T^{11} + 190128196 p^{8} T^{12} + 473152 p^{10} T^{13} + 2048 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \)
89 \( 1 - 40760 T^{2} + 852094356 T^{4} - 11466134726440 T^{6} + 107725553856705446 T^{8} - 11466134726440 p^{4} T^{10} + 852094356 p^{8} T^{12} - 40760 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 + 272 T + 36992 T^{2} + 4282960 T^{3} + 4431428 p T^{4} + 37180810864 T^{5} + 3384097431936 T^{6} + 328706617622448 T^{7} + 32062187105352070 T^{8} + 328706617622448 p^{2} T^{9} + 3384097431936 p^{4} T^{10} + 37180810864 p^{6} T^{11} + 4431428 p^{9} T^{12} + 4282960 p^{10} T^{13} + 36992 p^{12} T^{14} + 272 p^{14} T^{15} + p^{16} T^{16} \)
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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

−3.95408415441468099800470654092, −3.82197564350753191727200375160, −3.58029450445980521398017193279, −3.43702105999286963658219268888, −3.30045904930697423820925178136, −3.29660428050365296534939068369, −3.13537101973730384194378359734, −2.91287027929317081897136838536, −2.73655004573546611868548314149, −2.68624666211677724334210024954, −2.66175969289697407909667409084, −2.31006121171618667242043259749, −2.19100286421403794814699162810, −2.16242485265301082084770871683, −1.90601060653531026472496829166, −1.90092172207297677750823483593, −1.44542964164582666185956038866, −1.44310763590314684320989905387, −1.39156135656590926572776644340, −1.08370642359875662801880795558, −0.62905877907923150572258735944, −0.62123331105063167275318555110, −0.57516907752235252399628029504, −0.26802240943171695404419417785, −0.07839459542403006080630070633, 0.07839459542403006080630070633, 0.26802240943171695404419417785, 0.57516907752235252399628029504, 0.62123331105063167275318555110, 0.62905877907923150572258735944, 1.08370642359875662801880795558, 1.39156135656590926572776644340, 1.44310763590314684320989905387, 1.44542964164582666185956038866, 1.90092172207297677750823483593, 1.90601060653531026472496829166, 2.16242485265301082084770871683, 2.19100286421403794814699162810, 2.31006121171618667242043259749, 2.66175969289697407909667409084, 2.68624666211677724334210024954, 2.73655004573546611868548314149, 2.91287027929317081897136838536, 3.13537101973730384194378359734, 3.29660428050365296534939068369, 3.30045904930697423820925178136, 3.43702105999286963658219268888, 3.58029450445980521398017193279, 3.82197564350753191727200375160, 3.95408415441468099800470654092

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

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