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Properties

Label 16-5e16-1.1-c1e8-0-0
Degree 1616
Conductor 152587890625152587890625
Sign 11
Analytic cond. 2.52195×10−62.52195\times 10^{-6}
Root an. cond. 0.4467950.446795
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 5·3-s + 10·4-s + 25·6-s − 5·8-s + 10·9-s − 4·11-s − 50·12-s − 5·13-s − 16·16-s − 10·17-s − 50·18-s − 5·19-s + 20·22-s + 5·23-s + 25·24-s − 5·25-s + 25·26-s − 5·27-s − 5·29-s − 9·31-s + 30·32-s + 20·33-s + 50·34-s + 100·36-s + 30·37-s + 25·38-s + ⋯
L(s)  = 1  − 3.53·2-s − 2.88·3-s + 5·4-s + 10.2·6-s − 1.76·8-s + 10/3·9-s − 1.20·11-s − 14.4·12-s − 1.38·13-s − 4·16-s − 2.42·17-s − 11.7·18-s − 1.14·19-s + 4.26·22-s + 1.04·23-s + 5.10·24-s − 25-s + 4.90·26-s − 0.962·27-s − 0.928·29-s − 1.61·31-s + 5.30·32-s + 3.48·33-s + 8.57·34-s + 50/3·36-s + 4.93·37-s + 4.05·38-s + ⋯

Functional equation

Λ(s)=((516)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((516)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 5165^{16}
Sign: 11
Analytic conductor: 2.52195×10−62.52195\times 10^{-6}
Root analytic conductor: 0.4467950.446795
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 516, ( :[1/2]8), 1)(16,\ 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.0037351163720.003735116372
L(12)L(\frac12) ≈\approx 0.0037351163720.003735116372
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+pT2−4pT3+pT4−4p2T5+p3T6+p4T8 1 + p T^{2} - 4 p T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8}
good2 1+5T+15T2+15pT3+41T4+15pT5−5p2T6−55pT7−199T8−55p2T9−5p4T10+15p4T11+41p4T12+15p6T13+15p6T14+5p7T15+p8T16 1 + 5 T + 15 T^{2} + 15 p T^{3} + 41 T^{4} + 15 p T^{5} - 5 p^{2} T^{6} - 55 p T^{7} - 199 T^{8} - 55 p^{2} T^{9} - 5 p^{4} T^{10} + 15 p^{4} T^{11} + 41 p^{4} T^{12} + 15 p^{6} T^{13} + 15 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
3 1+5T+5pT2+10pT3+4p2T4+5T5−40pT6−400T7−809T8−400pT9−40p3T10+5p3T11+4p6T12+10p6T13+5p7T14+5p7T15+p8T16 1 + 5 T + 5 p T^{2} + 10 p T^{3} + 4 p^{2} T^{4} + 5 T^{5} - 40 p T^{6} - 400 T^{7} - 809 T^{8} - 400 p T^{9} - 40 p^{3} T^{10} + 5 p^{3} T^{11} + 4 p^{6} T^{12} + 10 p^{6} T^{13} + 5 p^{7} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
7 1−5pT2+611T4−7045T6+57976T8−7045p2T10+611p4T12−5p7T14+p8T16 1 - 5 p T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 7045 p^{2} T^{10} + 611 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16}
11 (1+2T−7T2−36T3+5T4−36pT5−7p2T6+2p3T7+p4T8)2 ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 36 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
13 1+5T+30T2+60T3+346T4+655T5+6335T6+13320T7+88856T8+13320pT9+6335p2T10+655p3T11+346p4T12+60p5T13+30p6T14+5p7T15+p8T16 1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 13320 p T^{9} + 6335 p^{2} T^{10} + 655 p^{3} T^{11} + 346 p^{4} T^{12} + 60 p^{5} T^{13} + 30 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
17 1+10T+90T2+720T3+4451T4+26310T5+136780T6+638720T7+2802941T8+638720pT9+136780p2T10+26310p3T11+4451p4T12+720p5T13+90p6T14+10p7T15+p8T16 1 + 10 T + 90 T^{2} + 720 T^{3} + 4451 T^{4} + 26310 T^{5} + 136780 T^{6} + 638720 T^{7} + 2802941 T^{8} + 638720 p T^{9} + 136780 p^{2} T^{10} + 26310 p^{3} T^{11} + 4451 p^{4} T^{12} + 720 p^{5} T^{13} + 90 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
19 1+5T−8T2+40T3+878T4+1705T5−1861T6+31550T7+293380T8+31550pT9−1861p2T10+1705p3T11+878p4T12+40p5T13−8p6T14+5p7T15+p8T16 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 31550 p T^{9} - 1861 p^{2} T^{10} + 1705 p^{3} T^{11} + 878 p^{4} T^{12} + 40 p^{5} T^{13} - 8 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
23 1−5T+45T2+100T3−104T4+7645T5+18890T6+8420T7+1238691T8+8420pT9+18890p2T10+7645p3T11−104p4T12+100p5T13+45p6T14−5p7T15+p8T16 1 - 5 T + 45 T^{2} + 100 T^{3} - 104 T^{4} + 7645 T^{5} + 18890 T^{6} + 8420 T^{7} + 1238691 T^{8} + 8420 p T^{9} + 18890 p^{2} T^{10} + 7645 p^{3} T^{11} - 104 p^{4} T^{12} + 100 p^{5} T^{13} + 45 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
29 1+5T−28T2+140T3+1268T4−5045T5+48219T6+207000T7−1738520T8+207000pT9+48219p2T10−5045p3T11+1268p4T12+140p5T13−28p6T14+5p7T15+p8T16 1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 207000 p T^{9} + 48219 p^{2} T^{10} - 5045 p^{3} T^{11} + 1268 p^{4} T^{12} + 140 p^{5} T^{13} - 28 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
31 1+9T+55T2+390T3+2980T4+20297T5+114748T6+589990T7+3582095T8+589990pT9+114748p2T10+20297p3T11+2980p4T12+390p5T13+55p6T14+9p7T15+p8T16 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 589990 p T^{9} + 114748 p^{2} T^{10} + 20297 p^{3} T^{11} + 2980 p^{4} T^{12} + 390 p^{5} T^{13} + 55 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16}
37 1−30T+480T2−5675T3+56171T4−489630T5+3826535T6−26904445T7+171416106T8−26904445pT9+3826535p2T10−489630p3T11+56171p4T12−5675p5T13+480p6T14−30p7T15+p8T16 1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 26904445 p T^{9} + 3826535 p^{2} T^{10} - 489630 p^{3} T^{11} + 56171 p^{4} T^{12} - 5675 p^{5} T^{13} + 480 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16}
41 1+4T−30T2−240T3+195T4+18372T5+77388T6−230240T7−1438395T8−230240pT9+77388p2T10+18372p3T11+195p4T12−240p5T13−30p6T14+4p7T15+p8T16 1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 230240 p T^{9} + 77388 p^{2} T^{10} + 18372 p^{3} T^{11} + 195 p^{4} T^{12} - 240 p^{5} T^{13} - 30 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
43 1−5pT2+22911T4−1578205T6+78597176T8−1578205p2T10+22911p4T12−5p7T14+p8T16 1 - 5 p T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16}
47 1+110T2−90T3+4101T4−9900T5−3830T6−910260T7−5610889T8−910260pT9−3830p2T10−9900p3T11+4101p4T12−90p5T13+110p6T14+p8T16 1 + 110 T^{2} - 90 T^{3} + 4101 T^{4} - 9900 T^{5} - 3830 T^{6} - 910260 T^{7} - 5610889 T^{8} - 910260 p T^{9} - 3830 p^{2} T^{10} - 9900 p^{3} T^{11} + 4101 p^{4} T^{12} - 90 p^{5} T^{13} + 110 p^{6} T^{14} + p^{8} T^{16}
53 1+10T+100T2+1625T3+11531T4+95310T5+857875T6+5635035T7+42472426T8+5635035pT9+857875p2T10+95310p3T11+11531p4T12+1625p5T13+100p6T14+10p7T15+p8T16 1 + 10 T + 100 T^{2} + 1625 T^{3} + 11531 T^{4} + 95310 T^{5} + 857875 T^{6} + 5635035 T^{7} + 42472426 T^{8} + 5635035 p T^{9} + 857875 p^{2} T^{10} + 95310 p^{3} T^{11} + 11531 p^{4} T^{12} + 1625 p^{5} T^{13} + 100 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
59 1−2pT2+900T3+3393T4−96300T5+615034T6+2943900T7−65890945T8+2943900pT9+615034p2T10−96300p3T11+3393p4T12+900p5T13−2p7T14+p8T16 1 - 2 p T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 2943900 p T^{9} + 615034 p^{2} T^{10} - 96300 p^{3} T^{11} + 3393 p^{4} T^{12} + 900 p^{5} T^{13} - 2 p^{7} T^{14} + p^{8} T^{16}
61 1+9T−165T2−1800T3+7560T4+154437T5+526138T6−4559670T7−69838275T8−4559670pT9+526138p2T10+154437p3T11+7560p4T12−1800p5T13−165p6T14+9p7T15+p8T16 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 4559670 p T^{9} + 526138 p^{2} T^{10} + 154437 p^{3} T^{11} + 7560 p^{4} T^{12} - 1800 p^{5} T^{13} - 165 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16}
67 1−20T+250T2−2600T3+26091T4−241820T5+2034700T6−15361680T7+117317461T8−15361680pT9+2034700p2T10−241820p3T11+26091p4T12−2600p5T13+250p6T14−20p7T15+p8T16 1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 15361680 p T^{9} + 2034700 p^{2} T^{10} - 241820 p^{3} T^{11} + 26091 p^{4} T^{12} - 2600 p^{5} T^{13} + 250 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16}
71 1−6T−910T3+4875T4+34402T5+474398T6−3835500T7−25760305T8−3835500pT9+474398p2T10+34402p3T11+4875p4T12−910p5T13−6p7T15+p8T16 1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 3835500 p T^{9} + 474398 p^{2} T^{10} + 34402 p^{3} T^{11} + 4875 p^{4} T^{12} - 910 p^{5} T^{13} - 6 p^{7} T^{15} + p^{8} T^{16}
73 1−15T+195T2−1340T3+15786T4−132465T5+1900340T6−200800pT7+154250461T8−200800p2T9+1900340p2T10−132465p3T11+15786p4T12−1340p5T13+195p6T14−15p7T15+p8T16 1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 200800 p T^{7} + 154250461 T^{8} - 200800 p^{2} T^{9} + 1900340 p^{2} T^{10} - 132465 p^{3} T^{11} + 15786 p^{4} T^{12} - 1340 p^{5} T^{13} + 195 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16}
79 1−15T−58T2+2180T3−7802T4−148865T5+1559169T6+5584500T7−169067020T8+5584500pT9+1559169p2T10−148865p3T11−7802p4T12+2180p5T13−58p6T14−15p7T15+p8T16 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 5584500 p T^{9} + 1559169 p^{2} T^{10} - 148865 p^{3} T^{11} - 7802 p^{4} T^{12} + 2180 p^{5} T^{13} - 58 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16}
83 1+45T+1115T2+19890T3+284376T4+3450645T5+448160pT6+367888080T7+3431240591T8+367888080pT9+448160p3T10+3450645p3T11+284376p4T12+19890p5T13+1115p6T14+45p7T15+p8T16 1 + 45 T + 1115 T^{2} + 19890 T^{3} + 284376 T^{4} + 3450645 T^{5} + 448160 p T^{6} + 367888080 T^{7} + 3431240591 T^{8} + 367888080 p T^{9} + 448160 p^{3} T^{10} + 3450645 p^{3} T^{11} + 284376 p^{4} T^{12} + 19890 p^{5} T^{13} + 1115 p^{6} T^{14} + 45 p^{7} T^{15} + p^{8} T^{16}
89 1+25T+342T2+5000T3+78868T4+928525T5+9098049T6+101226750T7+1076434080T8+101226750pT9+9098049p2T10+928525p3T11+78868p4T12+5000p5T13+342p6T14+25p7T15+p8T16 1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 101226750 p T^{9} + 9098049 p^{2} T^{10} + 928525 p^{3} T^{11} + 78868 p^{4} T^{12} + 5000 p^{5} T^{13} + 342 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16}
97 1+60T+1830T2+35660T3+467711T4+3770460T5+6583460T6−292271720T7−4451764059T8−292271720pT9+6583460p2T10+3770460p3T11+467711p4T12+35660p5T13+1830p6T14+60p7T15+p8T16 1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 292271720 p T^{9} + 6583460 p^{2} T^{10} + 3770460 p^{3} T^{11} + 467711 p^{4} T^{12} + 35660 p^{5} T^{13} + 1830 p^{6} T^{14} + 60 p^{7} T^{15} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(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

−9.264606005725786684581250817764, −8.701519957482883251445840148556, −8.700756789123480086234663064578, −8.587410407659643003356611381504, −8.313327649596639150128601111549, −8.022729140610343966777265765300, −8.000658794287396241187871279826, −7.73124663888318285965963720408, −7.37214446289042285892258381094, −7.31387626590244159416471837223, −7.04203274766215539700169588614, −6.85744330449983430802479407420, −6.54132635344714138363853998282, −6.45852707771919679444432037317, −5.82385696664051564495606249014, −5.66095007107135267912137327348, −5.61736961642789035495983217468, −5.52198619498244300122709572032, −4.97705277057159681438592943029, −4.73810456658168035860164939240, −4.36048904079619467511194010037, −4.20351592571445873243353432084, −3.98048819631383695223672152000, −2.51905553356217036605454282958, −2.48425883173590507807753867406, 2.48425883173590507807753867406, 2.51905553356217036605454282958, 3.98048819631383695223672152000, 4.20351592571445873243353432084, 4.36048904079619467511194010037, 4.73810456658168035860164939240, 4.97705277057159681438592943029, 5.52198619498244300122709572032, 5.61736961642789035495983217468, 5.66095007107135267912137327348, 5.82385696664051564495606249014, 6.45852707771919679444432037317, 6.54132635344714138363853998282, 6.85744330449983430802479407420, 7.04203274766215539700169588614, 7.31387626590244159416471837223, 7.37214446289042285892258381094, 7.73124663888318285965963720408, 8.000658794287396241187871279826, 8.022729140610343966777265765300, 8.313327649596639150128601111549, 8.587410407659643003356611381504, 8.700756789123480086234663064578, 8.701519957482883251445840148556, 9.264606005725786684581250817764

Graph of the ZZ-function along the critical line

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