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Properties

Label 16-26e8-1.1-c2e8-0-0
Degree 1616
Conductor 208827064576208827064576
Sign 11
Analytic cond. 0.06345510.0634551
Root an. cond. 0.8416930.841693
Motivic weight 22
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  − 4·2-s + 8·4-s + 6·5-s − 2·7-s − 16·8-s − 3·9-s − 24·10-s − 18·11-s + 36·13-s + 8·14-s + 36·16-s − 42·17-s + 12·18-s + 46·19-s + 48·20-s + 72·22-s − 36·23-s + 18·25-s − 144·26-s + 24·27-s − 16·28-s − 6·29-s + 32·31-s − 64·32-s + 168·34-s − 12·35-s − 24·36-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s + 6/5·5-s − 2/7·7-s − 2·8-s − 1/3·9-s − 2.39·10-s − 1.63·11-s + 2.76·13-s + 4/7·14-s + 9/4·16-s − 2.47·17-s + 2/3·18-s + 2.42·19-s + 12/5·20-s + 3.27·22-s − 1.56·23-s + 0.719·25-s − 5.53·26-s + 8/9·27-s − 4/7·28-s − 0.206·29-s + 1.03·31-s − 2·32-s + 4.94·34-s − 0.342·35-s − 2/3·36-s + ⋯

Functional equation

Λ(s)=((28⋅138)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((28⋅138)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅1382^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 0.06345510.0634551
Root analytic conductor: 0.8416930.841693
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅138, ( :[1]8), 1)(16,\ 2^{8} \cdot 13^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.24937133840.2493713384
L(12)L(\frac12) ≈\approx 0.24937133840.2493713384
L(2)L(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+pT+pT2+p2T3+p2T4)2 ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2}
13 1−36T+589T2−7824T3+8172pT4−7824p2T5+589p4T6−36p6T7+p8T8 1 - 36 T + 589 T^{2} - 7824 T^{3} + 8172 p T^{4} - 7824 p^{2} T^{5} + 589 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8}
good3 1+pT2−8pT3−29pT4+8p2T5−2p3T6+200p2T7+70p2T8+200p4T9−2p7T10+8p8T11−29p9T12−8p11T13+p13T14+p16T16 1 + p T^{2} - 8 p T^{3} - 29 p T^{4} + 8 p^{2} T^{5} - 2 p^{3} T^{6} + 200 p^{2} T^{7} + 70 p^{2} T^{8} + 200 p^{4} T^{9} - 2 p^{7} T^{10} + 8 p^{8} T^{11} - 29 p^{9} T^{12} - 8 p^{11} T^{13} + p^{13} T^{14} + p^{16} T^{16}
5 1−6T+18T2−48pT3+89pT4+2988T5+2862T6+36666T7−716556T8+36666p2T9+2862p4T10+2988p6T11+89p9T12−48p11T13+18p12T14−6p14T15+p16T16 1 - 6 T + 18 T^{2} - 48 p T^{3} + 89 p T^{4} + 2988 T^{5} + 2862 T^{6} + 36666 T^{7} - 716556 T^{8} + 36666 p^{2} T^{9} + 2862 p^{4} T^{10} + 2988 p^{6} T^{11} + 89 p^{9} T^{12} - 48 p^{11} T^{13} + 18 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16}
7 1+2T−127T2−62pT3+8255T4+22156T5−385248T6−453708T7+19028482T8−453708p2T9−385248p4T10+22156p6T11+8255p8T12−62p11T13−127p12T14+2p14T15+p16T16 1 + 2 T - 127 T^{2} - 62 p T^{3} + 8255 T^{4} + 22156 T^{5} - 385248 T^{6} - 453708 T^{7} + 19028482 T^{8} - 453708 p^{2} T^{9} - 385248 p^{4} T^{10} + 22156 p^{6} T^{11} + 8255 p^{8} T^{12} - 62 p^{11} T^{13} - 127 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16}
11 1+18T+105T2−3114T3−59393T4−565908T5+897792T6+64810956T7+1103342370T8+64810956p2T9+897792p4T10−565908p6T11−59393p8T12−3114p10T13+105p12T14+18p14T15+p16T16 1 + 18 T + 105 T^{2} - 3114 T^{3} - 59393 T^{4} - 565908 T^{5} + 897792 T^{6} + 64810956 T^{7} + 1103342370 T^{8} + 64810956 p^{2} T^{9} + 897792 p^{4} T^{10} - 565908 p^{6} T^{11} - 59393 p^{8} T^{12} - 3114 p^{10} T^{13} + 105 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16}
17 1+42T+1726T2+47796T3+1303465T4+30227796T5+646060522T6+12441458526T7+218725843324T8+12441458526p2T9+646060522p4T10+30227796p6T11+1303465p8T12+47796p10T13+1726p12T14+42p14T15+p16T16 1 + 42 T + 1726 T^{2} + 47796 T^{3} + 1303465 T^{4} + 30227796 T^{5} + 646060522 T^{6} + 12441458526 T^{7} + 218725843324 T^{8} + 12441458526 p^{2} T^{9} + 646060522 p^{4} T^{10} + 30227796 p^{6} T^{11} + 1303465 p^{8} T^{12} + 47796 p^{10} T^{13} + 1726 p^{12} T^{14} + 42 p^{14} T^{15} + p^{16} T^{16}
19 1−46T+977T2−19730T3+483311T4−8783612T5+6242688pT6−1680839364T7+28608116338T8−1680839364p2T9+6242688p5T10−8783612p6T11+483311p8T12−19730p10T13+977p12T14−46p14T15+p16T16 1 - 46 T + 977 T^{2} - 19730 T^{3} + 483311 T^{4} - 8783612 T^{5} + 6242688 p T^{6} - 1680839364 T^{7} + 28608116338 T^{8} - 1680839364 p^{2} T^{9} + 6242688 p^{5} T^{10} - 8783612 p^{6} T^{11} + 483311 p^{8} T^{12} - 19730 p^{10} T^{13} + 977 p^{12} T^{14} - 46 p^{14} T^{15} + p^{16} T^{16}
23 1+36T+2131T2+61164T3+2152429T4+44226000T5+1303450510T6+22452294672T7+648751140166T8+22452294672p2T9+1303450510p4T10+44226000p6T11+2152429p8T12+61164p10T13+2131p12T14+36p14T15+p16T16 1 + 36 T + 2131 T^{2} + 61164 T^{3} + 2152429 T^{4} + 44226000 T^{5} + 1303450510 T^{6} + 22452294672 T^{7} + 648751140166 T^{8} + 22452294672 p^{2} T^{9} + 1303450510 p^{4} T^{10} + 44226000 p^{6} T^{11} + 2152429 p^{8} T^{12} + 61164 p^{10} T^{13} + 2131 p^{12} T^{14} + 36 p^{14} T^{15} + p^{16} T^{16}
29 1+6T−2398T2−14772T3+2997253T4+14035500T5−3325399546T6−4535559294T7+3230740374436T8−4535559294p2T9−3325399546p4T10+14035500p6T11+2997253p8T12−14772p10T13−2398p12T14+6p14T15+p16T16 1 + 6 T - 2398 T^{2} - 14772 T^{3} + 2997253 T^{4} + 14035500 T^{5} - 3325399546 T^{6} - 4535559294 T^{7} + 3230740374436 T^{8} - 4535559294 p^{2} T^{9} - 3325399546 p^{4} T^{10} + 14035500 p^{6} T^{11} + 2997253 p^{8} T^{12} - 14772 p^{10} T^{13} - 2398 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16}
31 1−32T+512T2−24592T3+1881968T4−48317872T5+884987520T6−50977297056T7+2940963189598T8−50977297056p2T9+884987520p4T10−48317872p6T11+1881968p8T12−24592p10T13+512p12T14−32p14T15+p16T16 1 - 32 T + 512 T^{2} - 24592 T^{3} + 1881968 T^{4} - 48317872 T^{5} + 884987520 T^{6} - 50977297056 T^{7} + 2940963189598 T^{8} - 50977297056 p^{2} T^{9} + 884987520 p^{4} T^{10} - 48317872 p^{6} T^{11} + 1881968 p^{8} T^{12} - 24592 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16}
37 1+106T+8342T2+393068T3+15188585T4+276977564T5−1726711374T6−651456022674T7−28349233687076T8−651456022674p2T9−1726711374p4T10+276977564p6T11+15188585p8T12+393068p10T13+8342p12T14+106p14T15+p16T16 1 + 106 T + 8342 T^{2} + 393068 T^{3} + 15188585 T^{4} + 276977564 T^{5} - 1726711374 T^{6} - 651456022674 T^{7} - 28349233687076 T^{8} - 651456022674 p^{2} T^{9} - 1726711374 p^{4} T^{10} + 276977564 p^{6} T^{11} + 15188585 p^{8} T^{12} + 393068 p^{10} T^{13} + 8342 p^{12} T^{14} + 106 p^{14} T^{15} + p^{16} T^{16}
41 1−132T+10686T2−13512pT3+19988317T4−213687144T5−26569998546T6+2518966024308T7−124070043071700T8+2518966024308p2T9−26569998546p4T10−213687144p6T11+19988317p8T12−13512p11T13+10686p12T14−132p14T15+p16T16 1 - 132 T + 10686 T^{2} - 13512 p T^{3} + 19988317 T^{4} - 213687144 T^{5} - 26569998546 T^{6} + 2518966024308 T^{7} - 124070043071700 T^{8} + 2518966024308 p^{2} T^{9} - 26569998546 p^{4} T^{10} - 213687144 p^{6} T^{11} + 19988317 p^{8} T^{12} - 13512 p^{11} T^{13} + 10686 p^{12} T^{14} - 132 p^{14} T^{15} + p^{16} T^{16}
43 1+108T+253pT2+755028T3+48271669T4+2548436904T5+130030600102T6+5877290089008T7+263654368009582T8+5877290089008p2T9+130030600102p4T10+2548436904p6T11+48271669p8T12+755028p10T13+253p13T14+108p14T15+p16T16 1 + 108 T + 253 p T^{2} + 755028 T^{3} + 48271669 T^{4} + 2548436904 T^{5} + 130030600102 T^{6} + 5877290089008 T^{7} + 263654368009582 T^{8} + 5877290089008 p^{2} T^{9} + 130030600102 p^{4} T^{10} + 2548436904 p^{6} T^{11} + 48271669 p^{8} T^{12} + 755028 p^{10} T^{13} + 253 p^{13} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16}
47 1−60T+1800T2−154932T3+21951088T4−867532140T5+24541932312T6−2003115348324T7+163148147612766T8−2003115348324p2T9+24541932312p4T10−867532140p6T11+21951088p8T12−154932p10T13+1800p12T14−60p14T15+p16T16 1 - 60 T + 1800 T^{2} - 154932 T^{3} + 21951088 T^{4} - 867532140 T^{5} + 24541932312 T^{6} - 2003115348324 T^{7} + 163148147612766 T^{8} - 2003115348324 p^{2} T^{9} + 24541932312 p^{4} T^{10} - 867532140 p^{6} T^{11} + 21951088 p^{8} T^{12} - 154932 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16}
53 (1+66T+5917T2+184818T3+12226368T4+184818p2T5+5917p4T6+66p6T7+p8T8)2 ( 1 + 66 T + 5917 T^{2} + 184818 T^{3} + 12226368 T^{4} + 184818 p^{2} T^{5} + 5917 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} )^{2}
59 1−18T+5565T2−318294T3+23254759T4−1693659348T5+98133474288T6−8885846915100T7+327983542866810T8−8885846915100p2T9+98133474288p4T10−1693659348p6T11+23254759p8T12−318294p10T13+5565p12T14−18p14T15+p16T16 1 - 18 T + 5565 T^{2} - 318294 T^{3} + 23254759 T^{4} - 1693659348 T^{5} + 98133474288 T^{6} - 8885846915100 T^{7} + 327983542866810 T^{8} - 8885846915100 p^{2} T^{9} + 98133474288 p^{4} T^{10} - 1693659348 p^{6} T^{11} + 23254759 p^{8} T^{12} - 318294 p^{10} T^{13} + 5565 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16}
61 1−36T−11878T2+285336T3+88972381T4−1311266160T5−475974647662T6+1883345164380T7+2030181383859340T8+1883345164380p2T9−475974647662p4T10−1311266160p6T11+88972381p8T12+285336p10T13−11878p12T14−36p14T15+p16T16 1 - 36 T - 11878 T^{2} + 285336 T^{3} + 88972381 T^{4} - 1311266160 T^{5} - 475974647662 T^{6} + 1883345164380 T^{7} + 2030181383859340 T^{8} + 1883345164380 p^{2} T^{9} - 475974647662 p^{4} T^{10} - 1311266160 p^{6} T^{11} + 88972381 p^{8} T^{12} + 285336 p^{10} T^{13} - 11878 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16}
67 1+74T+15917T2+695782T3+97615703T4+493359916T5+227553513504T6−21342931471668T7+178591314167674T8−21342931471668p2T9+227553513504p4T10+493359916p6T11+97615703p8T12+695782p10T13+15917p12T14+74p14T15+p16T16 1 + 74 T + 15917 T^{2} + 695782 T^{3} + 97615703 T^{4} + 493359916 T^{5} + 227553513504 T^{6} - 21342931471668 T^{7} + 178591314167674 T^{8} - 21342931471668 p^{2} T^{9} + 227553513504 p^{4} T^{10} + 493359916 p^{6} T^{11} + 97615703 p^{8} T^{12} + 695782 p^{10} T^{13} + 15917 p^{12} T^{14} + 74 p^{14} T^{15} + p^{16} T^{16}
71 1+174T+14793T2+561450T3−2330369T4−584834196T5+83898109968T6+14951669162460T7+1250020614182514T8+14951669162460p2T9+83898109968p4T10−584834196p6T11−2330369p8T12+561450p10T13+14793p12T14+174p14T15+p16T16 1 + 174 T + 14793 T^{2} + 561450 T^{3} - 2330369 T^{4} - 584834196 T^{5} + 83898109968 T^{6} + 14951669162460 T^{7} + 1250020614182514 T^{8} + 14951669162460 p^{2} T^{9} + 83898109968 p^{4} T^{10} - 584834196 p^{6} T^{11} - 2330369 p^{8} T^{12} + 561450 p^{10} T^{13} + 14793 p^{12} T^{14} + 174 p^{14} T^{15} + p^{16} T^{16}
73 1−166T+13778T2−1402664T3+172017437T4−14047512068T5+945563904750T6−1139077533150pT7+1353169547476p2T8−1139077533150p3T9+945563904750p4T10−14047512068p6T11+172017437p8T12−1402664p10T13+13778p12T14−166p14T15+p16T16 1 - 166 T + 13778 T^{2} - 1402664 T^{3} + 172017437 T^{4} - 14047512068 T^{5} + 945563904750 T^{6} - 1139077533150 p T^{7} + 1353169547476 p^{2} T^{8} - 1139077533150 p^{3} T^{9} + 945563904750 p^{4} T^{10} - 14047512068 p^{6} T^{11} + 172017437 p^{8} T^{12} - 1402664 p^{10} T^{13} + 13778 p^{12} T^{14} - 166 p^{14} T^{15} + p^{16} T^{16}
79 (1+48T+10084T2+724368T3+50280774T4+724368p2T5+10084p4T6+48p6T7+p8T8)2 ( 1 + 48 T + 10084 T^{2} + 724368 T^{3} + 50280774 T^{4} + 724368 p^{2} T^{5} + 10084 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2}
83 1+240T+28800T2+2896512T3+298194640T4+338245056pT5+2344726766592T6+177350340390480T7+13560551099314782T8+177350340390480p2T9+2344726766592p4T10+338245056p7T11+298194640p8T12+2896512p10T13+28800p12T14+240p14T15+p16T16 1 + 240 T + 28800 T^{2} + 2896512 T^{3} + 298194640 T^{4} + 338245056 p T^{5} + 2344726766592 T^{6} + 177350340390480 T^{7} + 13560551099314782 T^{8} + 177350340390480 p^{2} T^{9} + 2344726766592 p^{4} T^{10} + 338245056 p^{7} T^{11} + 298194640 p^{8} T^{12} + 2896512 p^{10} T^{13} + 28800 p^{12} T^{14} + 240 p^{14} T^{15} + p^{16} T^{16}
89 1−294T+63735T2−10176306T3+1386303445T4−163006886868T5+17292939386730T6−1689166620371832T7+154185854226666234T8−1689166620371832p2T9+17292939386730p4T10−163006886868p6T11+1386303445p8T12−10176306p10T13+63735p12T14−294p14T15+p16T16 1 - 294 T + 63735 T^{2} - 10176306 T^{3} + 1386303445 T^{4} - 163006886868 T^{5} + 17292939386730 T^{6} - 1689166620371832 T^{7} + 154185854226666234 T^{8} - 1689166620371832 p^{2} T^{9} + 17292939386730 p^{4} T^{10} - 163006886868 p^{6} T^{11} + 1386303445 p^{8} T^{12} - 10176306 p^{10} T^{13} + 63735 p^{12} T^{14} - 294 p^{14} T^{15} + p^{16} T^{16}
97 1+58T+14363T2+998T3−4472611T4−1301881588T5−229071912798T6+178692170625312T7+7600464823934194T8+178692170625312p2T9−229071912798p4T10−1301881588p6T11−4472611p8T12+998p10T13+14363p12T14+58p14T15+p16T16 1 + 58 T + 14363 T^{2} + 998 T^{3} - 4472611 T^{4} - 1301881588 T^{5} - 229071912798 T^{6} + 178692170625312 T^{7} + 7600464823934194 T^{8} + 178692170625312 p^{2} T^{9} - 229071912798 p^{4} T^{10} - 1301881588 p^{6} T^{11} - 4472611 p^{8} T^{12} + 998 p^{10} T^{13} + 14363 p^{12} T^{14} + 58 p^{14} T^{15} + p^{16} 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

−8.508536336186558123739459273033, −8.415811805627628070892323830060, −8.414200846963753870410423786743, −8.408054940427337131476877541663, −7.938031072020424465250746126613, −7.51327801071358733594405478733, −7.32621940736378706558100165065, −7.19686918574269283431349104819, −7.00799684015701738839241920865, −6.64726784752073974216223813522, −6.59310787136014597901912063305, −6.07070787594747835733965990320, −5.86126712498175236756529226101, −5.79575268262008453379640960603, −5.68351322626094566961703263013, −5.59585237339799816418015853747, −4.73732082562542502302931571853, −4.67875794261568990535766948499, −4.51599376519828505595508846068, −3.72501728027450928871648736105, −3.34406627693961089623562420582, −3.32403950724458260739668169059, −2.45694893223072642811104559116, −2.39308500621309395742667048041, −1.40446580441834285903588825862, 1.40446580441834285903588825862, 2.39308500621309395742667048041, 2.45694893223072642811104559116, 3.32403950724458260739668169059, 3.34406627693961089623562420582, 3.72501728027450928871648736105, 4.51599376519828505595508846068, 4.67875794261568990535766948499, 4.73732082562542502302931571853, 5.59585237339799816418015853747, 5.68351322626094566961703263013, 5.79575268262008453379640960603, 5.86126712498175236756529226101, 6.07070787594747835733965990320, 6.59310787136014597901912063305, 6.64726784752073974216223813522, 7.00799684015701738839241920865, 7.19686918574269283431349104819, 7.32621940736378706558100165065, 7.51327801071358733594405478733, 7.938031072020424465250746126613, 8.408054940427337131476877541663, 8.414200846963753870410423786743, 8.415811805627628070892323830060, 8.508536336186558123739459273033

Graph of the ZZ-function along the critical line

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