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Properties

Label 16-220e8-1.1-c1e8-0-5
Degree 1616
Conductor 5.488×10185.488\times 10^{18}
Sign 11
Analytic cond. 90.698190.6981
Root an. cond. 1.325401.32540
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 + 12·4-s + 2·5-s + 25·6-s − 3·7-s + 20·8-s + 11·9-s + 10·10-s − 11-s + 60·12-s + 10·13-s − 15·14-s + 10·15-s + 30·16-s − 5·17-s + 55·18-s − 11·19-s + 24·20-s − 15·21-s − 5·22-s + 100·24-s + 25-s + 50·26-s + 5·27-s − 36·28-s − 5·29-s + ⋯
L(s)  = 1  + 3.53·2-s + 2.88·3-s + 6·4-s + 0.894·5-s + 10.2·6-s − 1.13·7-s + 7.07·8-s + 11/3·9-s + 3.16·10-s − 0.301·11-s + 17.3·12-s + 2.77·13-s − 4.00·14-s + 2.58·15-s + 15/2·16-s − 1.21·17-s + 12.9·18-s − 2.52·19-s + 5.36·20-s − 3.27·21-s − 1.06·22-s + 20.4·24-s + 1/5·25-s + 9.80·26-s + 0.962·27-s − 6.80·28-s − 0.928·29-s + ⋯

Functional equation

Λ(s)=((216⋅58⋅118)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅58⋅118)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅58⋅1182^{16} \cdot 5^{8} \cdot 11^{8}
Sign: 11
Analytic conductor: 90.698190.6981
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅58⋅118, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 48.2728628048.27286280
L(12)L(\frac12) ≈\approx 48.2728628048.27286280
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)
bad2 1−5T+13T2−25T3+39T4−25pT5+13p2T6−5p3T7+p4T8 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 25 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
5 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}
11 1+T+10T2−31T3−51T4−31pT5+10p2T6+p3T7+p4T8 1 + T + 10 T^{2} - 31 T^{3} - 51 T^{4} - 31 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
good3 1−5T+14T2−20T3+4pT4+5pT5−2p2T6−20T7+115T8−20pT9−2p4T10+5p4T11+4p5T12−20p5T13+14p6T14−5p7T15+p8T16 1 - 5 T + 14 T^{2} - 20 T^{3} + 4 p T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - 20 T^{7} + 115 T^{8} - 20 p T^{9} - 2 p^{4} T^{10} + 5 p^{4} T^{11} + 4 p^{5} T^{12} - 20 p^{5} T^{13} + 14 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
7 1+3T+9T2+23T3+33T4−16pT5−344T6−1866T7−6767T8−1866pT9−344p2T10−16p4T11+33p4T12+23p5T13+9p6T14+3p7T15+p8T16 1 + 3 T + 9 T^{2} + 23 T^{3} + 33 T^{4} - 16 p T^{5} - 344 T^{6} - 1866 T^{7} - 6767 T^{8} - 1866 p T^{9} - 344 p^{2} T^{10} - 16 p^{4} T^{11} + 33 p^{4} T^{12} + 23 p^{5} T^{13} + 9 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16}
13 1−10T+63T2−255T3+985T4−3725T5+17943T6−5880pT7+309609T8−5880p2T9+17943p2T10−3725p3T11+985p4T12−255p5T13+63p6T14−10p7T15+p8T16 1 - 10 T + 63 T^{2} - 255 T^{3} + 985 T^{4} - 3725 T^{5} + 17943 T^{6} - 5880 p T^{7} + 309609 T^{8} - 5880 p^{2} T^{9} + 17943 p^{2} T^{10} - 3725 p^{3} T^{11} + 985 p^{4} T^{12} - 255 p^{5} T^{13} + 63 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16}
17 1+5T+22T2+95T3+630T4+2570T5+11552T6+38480T7+157739T8+38480pT9+11552p2T10+2570p3T11+630p4T12+95p5T13+22p6T14+5p7T15+p8T16 1 + 5 T + 22 T^{2} + 95 T^{3} + 630 T^{4} + 2570 T^{5} + 11552 T^{6} + 38480 T^{7} + 157739 T^{8} + 38480 p T^{9} + 11552 p^{2} T^{10} + 2570 p^{3} T^{11} + 630 p^{4} T^{12} + 95 p^{5} T^{13} + 22 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
19 1+11T+pT2−104T3+281T4+4988T5+16586T6+34243T7+100777T8+34243pT9+16586p2T10+4988p3T11+281p4T12−104p5T13+p7T14+11p7T15+p8T16 1 + 11 T + p T^{2} - 104 T^{3} + 281 T^{4} + 4988 T^{5} + 16586 T^{6} + 34243 T^{7} + 100777 T^{8} + 34243 p T^{9} + 16586 p^{2} T^{10} + 4988 p^{3} T^{11} + 281 p^{4} T^{12} - 104 p^{5} T^{13} + p^{7} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16}
23 1−3pT2+2652T4−73647T6+1794335T8−73647p2T10+2652p4T12−3p7T14+p8T16 1 - 3 p T^{2} + 2652 T^{4} - 73647 T^{6} + 1794335 T^{8} - 73647 p^{2} T^{10} + 2652 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16}
29 1+5T+65T2+390T3+101pT4+16620T5+100910T6+18895pT7+94619pT8+18895p2T9+100910p2T10+16620p3T11+101p5T12+390p5T13+65p6T14+5p7T15+p8T16 1 + 5 T + 65 T^{2} + 390 T^{3} + 101 p T^{4} + 16620 T^{5} + 100910 T^{6} + 18895 p T^{7} + 94619 p T^{8} + 18895 p^{2} T^{9} + 100910 p^{2} T^{10} + 16620 p^{3} T^{11} + 101 p^{5} T^{12} + 390 p^{5} T^{13} + 65 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
31 1−10T+125T2−1025T3+7269T4−47065T5+8105pT6−1412070T7+7458211T8−1412070pT9+8105p3T10−47065p3T11+7269p4T12−1025p5T13+125p6T14−10p7T15+p8T16 1 - 10 T + 125 T^{2} - 1025 T^{3} + 7269 T^{4} - 47065 T^{5} + 8105 p T^{6} - 1412070 T^{7} + 7458211 T^{8} - 1412070 p T^{9} + 8105 p^{3} T^{10} - 47065 p^{3} T^{11} + 7269 p^{4} T^{12} - 1025 p^{5} T^{13} + 125 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16}
37 1−20T+121T2−100T3+422T4−24080T5+162413T6−352450T7−62645T8−352450pT9+162413p2T10−24080p3T11+422p4T12−100p5T13+121p6T14−20p7T15+p8T16 1 - 20 T + 121 T^{2} - 100 T^{3} + 422 T^{4} - 24080 T^{5} + 162413 T^{6} - 352450 T^{7} - 62645 T^{8} - 352450 p T^{9} + 162413 p^{2} T^{10} - 24080 p^{3} T^{11} + 422 p^{4} T^{12} - 100 p^{5} T^{13} + 121 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16}
41 1−10T+139T2−790T3+4350T4−100T5−182029T6+1962350T7−16190221T8+1962350pT9−182029p2T10−100p3T11+4350p4T12−790p5T13+139p6T14−10p7T15+p8T16 1 - 10 T + 139 T^{2} - 790 T^{3} + 4350 T^{4} - 100 T^{5} - 182029 T^{6} + 1962350 T^{7} - 16190221 T^{8} + 1962350 p T^{9} - 182029 p^{2} T^{10} - 100 p^{3} T^{11} + 4350 p^{4} T^{12} - 790 p^{5} T^{13} + 139 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16}
43 (1+17T+261T2+2356T3+18809T4+2356pT5+261p2T6+17p3T7+p4T8)2 ( 1 + 17 T + 261 T^{2} + 2356 T^{3} + 18809 T^{4} + 2356 p T^{5} + 261 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 1+79T2−615T3+4647T4−48585T5+415847T6−2255820T7+23961215T8−2255820pT9+415847p2T10−48585p3T11+4647p4T12−615p5T13+79p6T14+p8T16 1 + 79 T^{2} - 615 T^{3} + 4647 T^{4} - 48585 T^{5} + 415847 T^{6} - 2255820 T^{7} + 23961215 T^{8} - 2255820 p T^{9} + 415847 p^{2} T^{10} - 48585 p^{3} T^{11} + 4647 p^{4} T^{12} - 615 p^{5} T^{13} + 79 p^{6} T^{14} + p^{8} T^{16}
53 1+7T−63T2−699T3−1039T4+60732T5+451460T6−1646222T7−29558181T8−1646222pT9+451460p2T10+60732p3T11−1039p4T12−699p5T13−63p6T14+7p7T15+p8T16 1 + 7 T - 63 T^{2} - 699 T^{3} - 1039 T^{4} + 60732 T^{5} + 451460 T^{6} - 1646222 T^{7} - 29558181 T^{8} - 1646222 p T^{9} + 451460 p^{2} T^{10} + 60732 p^{3} T^{11} - 1039 p^{4} T^{12} - 699 p^{5} T^{13} - 63 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16}
59 1+5T+55T2+1055T3+7269T4+59120T5+675980T6+4611020T7+35189921T8+4611020pT9+675980p2T10+59120p3T11+7269p4T12+1055p5T13+55p6T14+5p7T15+p8T16 1 + 5 T + 55 T^{2} + 1055 T^{3} + 7269 T^{4} + 59120 T^{5} + 675980 T^{6} + 4611020 T^{7} + 35189921 T^{8} + 4611020 p T^{9} + 675980 p^{2} T^{10} + 59120 p^{3} T^{11} + 7269 p^{4} T^{12} + 1055 p^{5} T^{13} + 55 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
61 1+10T+40T2−970T3−11951T4−57310T5+116720T6+4738300T7+39395361T8+4738300pT9+116720p2T10−57310p3T11−11951p4T12−970p5T13+40p6T14+10p7T15+p8T16 1 + 10 T + 40 T^{2} - 970 T^{3} - 11951 T^{4} - 57310 T^{5} + 116720 T^{6} + 4738300 T^{7} + 39395361 T^{8} + 4738300 p T^{9} + 116720 p^{2} T^{10} - 57310 p^{3} T^{11} - 11951 p^{4} T^{12} - 970 p^{5} T^{13} + 40 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
67 1−349T2+59052T4−6537947T6+515546375T8−6537947p2T10+59052p4T12−349p6T14+p8T16 1 - 349 T^{2} + 59052 T^{4} - 6537947 T^{6} + 515546375 T^{8} - 6537947 p^{2} T^{10} + 59052 p^{4} T^{12} - 349 p^{6} T^{14} + p^{8} T^{16}
71 1−90T+3925T2−110820T3+2283354T4−36667110T5+477857375T6−5182638450T7+47432387951T8−5182638450pT9+477857375p2T10−36667110p3T11+2283354p4T12−110820p5T13+3925p6T14−90p7T15+p8T16 1 - 90 T + 3925 T^{2} - 110820 T^{3} + 2283354 T^{4} - 36667110 T^{5} + 477857375 T^{6} - 5182638450 T^{7} + 47432387951 T^{8} - 5182638450 p T^{9} + 477857375 p^{2} T^{10} - 36667110 p^{3} T^{11} + 2283354 p^{4} T^{12} - 110820 p^{5} T^{13} + 3925 p^{6} T^{14} - 90 p^{7} T^{15} + p^{8} T^{16}
73 1−5T+248T2−805T3+20520T4−4970T5+425438T6+6710670T7−20435801T8+6710670pT9+425438p2T10−4970p3T11+20520p4T12−805p5T13+248p6T14−5p7T15+p8T16 1 - 5 T + 248 T^{2} - 805 T^{3} + 20520 T^{4} - 4970 T^{5} + 425438 T^{6} + 6710670 T^{7} - 20435801 T^{8} + 6710670 p T^{9} + 425438 p^{2} T^{10} - 4970 p^{3} T^{11} + 20520 p^{4} T^{12} - 805 p^{5} T^{13} + 248 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
79 1+21T+184T2+1056T3+9486T4+95943T5+278336T6−4558662T7−59027833T8−4558662pT9+278336p2T10+95943p3T11+9486p4T12+1056p5T13+184p6T14+21p7T15+p8T16 1 + 21 T + 184 T^{2} + 1056 T^{3} + 9486 T^{4} + 95943 T^{5} + 278336 T^{6} - 4558662 T^{7} - 59027833 T^{8} - 4558662 p T^{9} + 278336 p^{2} T^{10} + 95943 p^{3} T^{11} + 9486 p^{4} T^{12} + 1056 p^{5} T^{13} + 184 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16}
83 1−19T−49T2+4489T3−30697T4−331754T5+5033994T6+7836182T7−444554067T8+7836182pT9+5033994p2T10−331754p3T11−30697p4T12+4489p5T13−49p6T14−19p7T15+p8T16 1 - 19 T - 49 T^{2} + 4489 T^{3} - 30697 T^{4} - 331754 T^{5} + 5033994 T^{6} + 7836182 T^{7} - 444554067 T^{8} + 7836182 p T^{9} + 5033994 p^{2} T^{10} - 331754 p^{3} T^{11} - 30697 p^{4} T^{12} + 4489 p^{5} T^{13} - 49 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16}
89 (1−10T+236T2−1540T3+26461T4−1540pT5+236p2T6−10p3T7+p4T8)2 ( 1 - 10 T + 236 T^{2} - 1540 T^{3} + 26461 T^{4} - 1540 p T^{5} + 236 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 1+44T+713T2+4508T3−6314T4−300796T5−1224575T6+43495854T7+786406539T8+43495854pT9−1224575p2T10−300796p3T11−6314p4T12+4508p5T13+713p6T14+44p7T15+p8T16 1 + 44 T + 713 T^{2} + 4508 T^{3} - 6314 T^{4} - 300796 T^{5} - 1224575 T^{6} + 43495854 T^{7} + 786406539 T^{8} + 43495854 p T^{9} - 1224575 p^{2} T^{10} - 300796 p^{3} T^{11} - 6314 p^{4} T^{12} + 4508 p^{5} T^{13} + 713 p^{6} T^{14} + 44 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

−5.61612435607151835289827886636, −5.47427156556099582481359172384, −5.22573931640927602183731235124, −4.96130055507239827965535109454, −4.84581392550468243132209204006, −4.66955891635857849735655446844, −4.61954457434242279951035309727, −4.48924300063501911734850840226, −4.19752633136315987590021497591, −3.96100825597302374629343378617, −3.83880409490315311350009556010, −3.73459298300824883293314798711, −3.69470324510167727779705716799, −3.65812977026697535832247971015, −3.37106599445265061565678770946, −3.24057716808216265642205423763, −2.85893416084571325850330162329, −2.74984404934146479115254744435, −2.53953284929822799054687769771, −2.47123309731214470634783270092, −2.26720114673310848632125787792, −1.94965721694096998715290383455, −1.68325905157608750826784759236, −1.62907234149835594352884690199, −0.964619208924802567703869705844, 0.964619208924802567703869705844, 1.62907234149835594352884690199, 1.68325905157608750826784759236, 1.94965721694096998715290383455, 2.26720114673310848632125787792, 2.47123309731214470634783270092, 2.53953284929822799054687769771, 2.74984404934146479115254744435, 2.85893416084571325850330162329, 3.24057716808216265642205423763, 3.37106599445265061565678770946, 3.65812977026697535832247971015, 3.69470324510167727779705716799, 3.73459298300824883293314798711, 3.83880409490315311350009556010, 3.96100825597302374629343378617, 4.19752633136315987590021497591, 4.48924300063501911734850840226, 4.61954457434242279951035309727, 4.66955891635857849735655446844, 4.84581392550468243132209204006, 4.96130055507239827965535109454, 5.22573931640927602183731235124, 5.47427156556099582481359172384, 5.61612435607151835289827886636

Graph of the ZZ-function along the critical line

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