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Properties

Label 16-2548e8-1.1-c1e8-0-1
Degree 1616
Conductor 1.777×10271.777\times 10^{27}
Sign 11
Analytic cond. 2.93637×10102.93637\times 10^{10}
Root an. cond. 4.510644.51064
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s + 6·9-s − 4·11-s + 8·13-s − 3·15-s − 11·23-s + 8·25-s − 3·27-s + 22·29-s − 5·31-s − 12·33-s − 27·37-s + 24·39-s − 12·41-s + 8·43-s − 6·45-s + 4·47-s − 18·53-s + 4·55-s − 16·59-s + 10·61-s − 8·65-s + 2·67-s − 33·69-s − 10·71-s − 7·73-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 2·9-s − 1.20·11-s + 2.21·13-s − 0.774·15-s − 2.29·23-s + 8/5·25-s − 0.577·27-s + 4.08·29-s − 0.898·31-s − 2.08·33-s − 4.43·37-s + 3.84·39-s − 1.87·41-s + 1.21·43-s − 0.894·45-s + 0.583·47-s − 2.47·53-s + 0.539·55-s − 2.08·59-s + 1.28·61-s − 0.992·65-s + 0.244·67-s − 3.97·69-s − 1.18·71-s − 0.819·73-s + ⋯

Functional equation

Λ(s)=((216⋅716⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅716⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅716⋅1382^{16} \cdot 7^{16} \cdot 13^{8}
Sign: 11
Analytic conductor: 2.93637×10102.93637\times 10^{10}
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅716⋅138, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 7^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.7918717061.791871706
L(12)L(\frac12) ≈\approx 1.7918717061.791871706
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 1
7 1 1
13 (1−T)8 ( 1 - T )^{8}
good3 (1−pT+pT2+2pT3−8pT4+2p2T5+p3T6−p4T7+p4T8)(1+2pT3+pT4+2p2T5+p4T8) ( 1 - p T + p T^{2} + 2 p T^{3} - 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + 2 p T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{4} T^{8} )
5 1+T−7T2−6T3+11T4−T5+46T6+87T7−104T8+87pT9+46p2T10−p3T11+11p4T12−6p5T13−7p6T14+p7T15+p8T16 1 + T - 7 T^{2} - 6 T^{3} + 11 T^{4} - T^{5} + 46 T^{6} + 87 T^{7} - 104 T^{8} + 87 p T^{9} + 46 p^{2} T^{10} - p^{3} T^{11} + 11 p^{4} T^{12} - 6 p^{5} T^{13} - 7 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16}
11 1+4T−10T2+36T3+311T4−574T5−2T6+6246T7−14120T8+6246pT9−2p2T10−574p3T11+311p4T12+36p5T13−10p6T14+4p7T15+p8T16 1 + 4 T - 10 T^{2} + 36 T^{3} + 311 T^{4} - 574 T^{5} - 2 T^{6} + 6246 T^{7} - 14120 T^{8} + 6246 p T^{9} - 2 p^{2} T^{10} - 574 p^{3} T^{11} + 311 p^{4} T^{12} + 36 p^{5} T^{13} - 10 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
17 1−26T2+120T3+277T4−2580T5+8254T6+29100T7−180884T8+29100pT9+8254p2T10−2580p3T11+277p4T12+120p5T13−26p6T14+p8T16 1 - 26 T^{2} + 120 T^{3} + 277 T^{4} - 2580 T^{5} + 8254 T^{6} + 29100 T^{7} - 180884 T^{8} + 29100 p T^{9} + 8254 p^{2} T^{10} - 2580 p^{3} T^{11} + 277 p^{4} T^{12} + 120 p^{5} T^{13} - 26 p^{6} T^{14} + p^{8} T^{16}
19 1−64T2+12T3+2365T4−498T5−64540T6+4206T7+1386700T8+4206pT9−64540p2T10−498p3T11+2365p4T12+12p5T13−64p6T14+p8T16 1 - 64 T^{2} + 12 T^{3} + 2365 T^{4} - 498 T^{5} - 64540 T^{6} + 4206 T^{7} + 1386700 T^{8} + 4206 p T^{9} - 64540 p^{2} T^{10} - 498 p^{3} T^{11} + 2365 p^{4} T^{12} + 12 p^{5} T^{13} - 64 p^{6} T^{14} + p^{8} T^{16}
23 1+11T−7T2−174T3+2849T4+13093T5−57770T6+44955T7+2974330T8+44955pT9−57770p2T10+13093p3T11+2849p4T12−174p5T13−7p6T14+11p7T15+p8T16 1 + 11 T - 7 T^{2} - 174 T^{3} + 2849 T^{4} + 13093 T^{5} - 57770 T^{6} + 44955 T^{7} + 2974330 T^{8} + 44955 p T^{9} - 57770 p^{2} T^{10} + 13093 p^{3} T^{11} + 2849 p^{4} T^{12} - 174 p^{5} T^{13} - 7 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16}
29 (1−11T+89T2−506T3+2734T4−506pT5+89p2T6−11p3T7+p4T8)2 ( 1 - 11 T + 89 T^{2} - 506 T^{3} + 2734 T^{4} - 506 p T^{5} + 89 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2}
31 1+5T−33T2−608T3−823T4+20541T5+130846T6−315001T7−5145042T8−315001pT9+130846p2T10+20541p3T11−823p4T12−608p5T13−33p6T14+5p7T15+p8T16 1 + 5 T - 33 T^{2} - 608 T^{3} - 823 T^{4} + 20541 T^{5} + 130846 T^{6} - 315001 T^{7} - 5145042 T^{8} - 315001 p T^{9} + 130846 p^{2} T^{10} + 20541 p^{3} T^{11} - 823 p^{4} T^{12} - 608 p^{5} T^{13} - 33 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
37 1+27T+317T2+2922T3+29599T4+252777T5+1729142T6+12208317T7+82743664T8+12208317pT9+1729142p2T10+252777p3T11+29599p4T12+2922p5T13+317p6T14+27p7T15+p8T16 1 + 27 T + 317 T^{2} + 2922 T^{3} + 29599 T^{4} + 252777 T^{5} + 1729142 T^{6} + 12208317 T^{7} + 82743664 T^{8} + 12208317 p T^{9} + 1729142 p^{2} T^{10} + 252777 p^{3} T^{11} + 29599 p^{4} T^{12} + 2922 p^{5} T^{13} + 317 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16}
41 (1+6T+4pT2+702T3+10062T4+702pT5+4p3T6+6p3T7+p4T8)2 ( 1 + 6 T + 4 p T^{2} + 702 T^{3} + 10062 T^{4} + 702 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 (1−4T+148T2−496T3+9046T4−496pT5+148p2T6−4p3T7+p4T8)2 ( 1 - 4 T + 148 T^{2} - 496 T^{3} + 9046 T^{4} - 496 p T^{5} + 148 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 1−4T−118T2−132T3+9551T4+24214T5−423158T6−589518T7+14574472T8−589518pT9−423158p2T10+24214p3T11+9551p4T12−132p5T13−118p6T14−4p7T15+p8T16 1 - 4 T - 118 T^{2} - 132 T^{3} + 9551 T^{4} + 24214 T^{5} - 423158 T^{6} - 589518 T^{7} + 14574472 T^{8} - 589518 p T^{9} - 423158 p^{2} T^{10} + 24214 p^{3} T^{11} + 9551 p^{4} T^{12} - 132 p^{5} T^{13} - 118 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
53 1+18T+40T2−192T3+9691T4+57480T5−498428T6−1193106T7+28851040T8−1193106pT9−498428p2T10+57480p3T11+9691p4T12−192p5T13+40p6T14+18p7T15+p8T16 1 + 18 T + 40 T^{2} - 192 T^{3} + 9691 T^{4} + 57480 T^{5} - 498428 T^{6} - 1193106 T^{7} + 28851040 T^{8} - 1193106 p T^{9} - 498428 p^{2} T^{10} + 57480 p^{3} T^{11} + 9691 p^{4} T^{12} - 192 p^{5} T^{13} + 40 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16}
59 1+16T+26T2−252T3+3803T4+3662T5−446918T6−43710pT7−134704pT8−43710p2T9−446918p2T10+3662p3T11+3803p4T12−252p5T13+26p6T14+16p7T15+p8T16 1 + 16 T + 26 T^{2} - 252 T^{3} + 3803 T^{4} + 3662 T^{5} - 446918 T^{6} - 43710 p T^{7} - 134704 p T^{8} - 43710 p^{2} T^{9} - 446918 p^{2} T^{10} + 3662 p^{3} T^{11} + 3803 p^{4} T^{12} - 252 p^{5} T^{13} + 26 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16}
61 1−10T−24T2+160T3−37T4+38760T5−57404T6−2183350T7+15224832T8−2183350pT9−57404p2T10+38760p3T11−37p4T12+160p5T13−24p6T14−10p7T15+p8T16 1 - 10 T - 24 T^{2} + 160 T^{3} - 37 T^{4} + 38760 T^{5} - 57404 T^{6} - 2183350 T^{7} + 15224832 T^{8} - 2183350 p T^{9} - 57404 p^{2} T^{10} + 38760 p^{3} T^{11} - 37 p^{4} T^{12} + 160 p^{5} T^{13} - 24 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16}
67 1−2T−138T2+104T3+9779T4−558T5−133058T6−124388T7−10842816T8−124388pT9−133058p2T10−558p3T11+9779p4T12+104p5T13−138p6T14−2p7T15+p8T16 1 - 2 T - 138 T^{2} + 104 T^{3} + 9779 T^{4} - 558 T^{5} - 133058 T^{6} - 124388 T^{7} - 10842816 T^{8} - 124388 p T^{9} - 133058 p^{2} T^{10} - 558 p^{3} T^{11} + 9779 p^{4} T^{12} + 104 p^{5} T^{13} - 138 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
71 (1+5T+17T2+200T3+4540T4+200pT5+17p2T6+5p3T7+p4T8)2 ( 1 + 5 T + 17 T^{2} + 200 T^{3} + 4540 T^{4} + 200 p T^{5} + 17 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 1+7T−93T2+1088T3+14315T4−96597T5+662386T6+8704183T7−59722104T8+8704183pT9+662386p2T10−96597p3T11+14315p4T12+1088p5T13−93p6T14+7p7T15+p8T16 1 + 7 T - 93 T^{2} + 1088 T^{3} + 14315 T^{4} - 96597 T^{5} + 662386 T^{6} + 8704183 T^{7} - 59722104 T^{8} + 8704183 p T^{9} + 662386 p^{2} T^{10} - 96597 p^{3} T^{11} + 14315 p^{4} T^{12} + 1088 p^{5} T^{13} - 93 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16}
79 1+17T+15T2−1664T3−10375T4+37557T5+477130T6+854183T7−6106818T8+854183pT9+477130p2T10+37557p3T11−10375p4T12−1664p5T13+15p6T14+17p7T15+p8T16 1 + 17 T + 15 T^{2} - 1664 T^{3} - 10375 T^{4} + 37557 T^{5} + 477130 T^{6} + 854183 T^{7} - 6106818 T^{8} + 854183 p T^{9} + 477130 p^{2} T^{10} + 37557 p^{3} T^{11} - 10375 p^{4} T^{12} - 1664 p^{5} T^{13} + 15 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16}
83 (1−6T+224T2−1386T3+24702T4−1386pT5+224p2T6−6p3T7+p4T8)2 ( 1 - 6 T + 224 T^{2} - 1386 T^{3} + 24702 T^{4} - 1386 p T^{5} + 224 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
89 1+21T+205T2+3774T3+44467T4+241719T5+3262366T6+28535535T7+88183516T8+28535535pT9+3262366p2T10+241719p3T11+44467p4T12+3774p5T13+205p6T14+21p7T15+p8T16 1 + 21 T + 205 T^{2} + 3774 T^{3} + 44467 T^{4} + 241719 T^{5} + 3262366 T^{6} + 28535535 T^{7} + 88183516 T^{8} + 28535535 p T^{9} + 3262366 p^{2} T^{10} + 241719 p^{3} T^{11} + 44467 p^{4} T^{12} + 3774 p^{5} T^{13} + 205 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16}
97 (1+8T+340T2+1916T3+46894T4+1916pT5+340p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 340 T^{2} + 1916 T^{3} + 46894 T^{4} + 1916 p T^{5} + 340 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
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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

−3.61914198579697736177036990906, −3.58416118141684586700075081828, −3.54950848688825823330494197906, −3.45552658800052575506872833807, −3.34239639308767275478762647485, −3.05905441677034375387855924662, −2.98364858458576681767738987899, −2.81245077354064279671794595459, −2.76519662881287111236824690571, −2.67629471545837366840679468822, −2.58364410197721044707594855874, −2.36473968577114826065690033881, −2.34543182922508430016804538687, −1.90171229427781406769804936287, −1.82020205940817116935194548616, −1.76386360523683778367714951026, −1.73905288553886228804944013450, −1.59695615221545556212158573819, −1.39136273582685947586855877262, −1.28282915371498104067825995852, −1.23899854170823181216399056247, −0.72042477112112005480562996158, −0.50900994753450451069204247102, −0.48779948226781878516914523499, −0.088413746940213388209939123855, 0.088413746940213388209939123855, 0.48779948226781878516914523499, 0.50900994753450451069204247102, 0.72042477112112005480562996158, 1.23899854170823181216399056247, 1.28282915371498104067825995852, 1.39136273582685947586855877262, 1.59695615221545556212158573819, 1.73905288553886228804944013450, 1.76386360523683778367714951026, 1.82020205940817116935194548616, 1.90171229427781406769804936287, 2.34543182922508430016804538687, 2.36473968577114826065690033881, 2.58364410197721044707594855874, 2.67629471545837366840679468822, 2.76519662881287111236824690571, 2.81245077354064279671794595459, 2.98364858458576681767738987899, 3.05905441677034375387855924662, 3.34239639308767275478762647485, 3.45552658800052575506872833807, 3.54950848688825823330494197906, 3.58416118141684586700075081828, 3.61914198579697736177036990906

Graph of the ZZ-function along the critical line

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