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Properties

Label 16-960e8-1.1-c2e8-0-9
Degree 1616
Conductor 7.214×10237.214\times 10^{23}
Sign 11
Analytic cond. 2.19204×10112.19204\times 10^{11}
Root an. cond. 5.114495.11449
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  + 12·5-s − 4·7-s + 32·11-s + 4·13-s + 52·17-s + 40·23-s + 30·25-s − 96·31-s − 48·35-s + 60·37-s − 152·41-s − 88·43-s + 16·47-s + 8·49-s − 108·53-s + 384·55-s − 264·61-s + 48·65-s − 216·67-s + 240·71-s − 208·73-s − 128·77-s − 18·81-s + 336·83-s + 624·85-s − 16·91-s − 208·97-s + ⋯
L(s)  = 1  + 12/5·5-s − 4/7·7-s + 2.90·11-s + 4/13·13-s + 3.05·17-s + 1.73·23-s + 6/5·25-s − 3.09·31-s − 1.37·35-s + 1.62·37-s − 3.70·41-s − 2.04·43-s + 0.340·47-s + 8/49·49-s − 2.03·53-s + 6.98·55-s − 4.32·61-s + 0.738·65-s − 3.22·67-s + 3.38·71-s − 2.84·73-s − 1.66·77-s − 2/9·81-s + 4.04·83-s + 7.34·85-s − 0.175·91-s − 2.14·97-s + ⋯

Functional equation

Λ(s)=((248⋅38⋅58)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((248⋅38⋅58)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 248⋅38⋅582^{48} \cdot 3^{8} \cdot 5^{8}
Sign: 11
Analytic conductor: 2.19204×10112.19204\times 10^{11}
Root analytic conductor: 5.114495.11449
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 248⋅38⋅58, ( :[1]8), 1)(16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 12.3734974412.37349744
L(12)L(\frac12) ≈\approx 12.3734974412.37349744
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 1
3 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
5 1−12T+114T2−708T3+826pT4−708p2T5+114p4T6−12p6T7+p8T8 1 - 12 T + 114 T^{2} - 708 T^{3} + 826 p T^{4} - 708 p^{2} T^{5} + 114 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8}
good7 1+4T+8T2+316T3−2816T4−2108T5+64024T6+120828T7+11331966T8+120828p2T9+64024p4T10−2108p6T11−2816p8T12+316p10T13+8p12T14+4p14T15+p16T16 1 + 4 T + 8 T^{2} + 316 T^{3} - 2816 T^{4} - 2108 T^{5} + 64024 T^{6} + 120828 T^{7} + 11331966 T^{8} + 120828 p^{2} T^{9} + 64024 p^{4} T^{10} - 2108 p^{6} T^{11} - 2816 p^{8} T^{12} + 316 p^{10} T^{13} + 8 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16}
11 (1−16T+126T2+1888T3−31078T4+1888p2T5+126p4T6−16p6T7+p8T8)2 ( 1 - 16 T + 126 T^{2} + 1888 T^{3} - 31078 T^{4} + 1888 p^{2} T^{5} + 126 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2}
13 1−4T+8T2−124pT3+1072pT4+328076T5−124520T6+28503876T7−763526370T8+28503876p2T9−124520p4T10+328076p6T11+1072p9T12−124p11T13+8p12T14−4p14T15+p16T16 1 - 4 T + 8 T^{2} - 124 p T^{3} + 1072 p T^{4} + 328076 T^{5} - 124520 T^{6} + 28503876 T^{7} - 763526370 T^{8} + 28503876 p^{2} T^{9} - 124520 p^{4} T^{10} + 328076 p^{6} T^{11} + 1072 p^{9} T^{12} - 124 p^{11} T^{13} + 8 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16}
17 1−52T+1352T2−28204T3+490768T4−6510964T5+72784600T6−481409004T7+154737438T8−481409004p2T9+72784600p4T10−6510964p6T11+490768p8T12−28204p10T13+1352p12T14−52p14T15+p16T16 1 - 52 T + 1352 T^{2} - 28204 T^{3} + 490768 T^{4} - 6510964 T^{5} + 72784600 T^{6} - 481409004 T^{7} + 154737438 T^{8} - 481409004 p^{2} T^{9} + 72784600 p^{4} T^{10} - 6510964 p^{6} T^{11} + 490768 p^{8} T^{12} - 28204 p^{10} T^{13} + 1352 p^{12} T^{14} - 52 p^{14} T^{15} + p^{16} T^{16}
19 1−2000T2+1889820T4−1128357424T6+476406585350T8−1128357424p4T10+1889820p8T12−2000p12T14+p16T16 1 - 2000 T^{2} + 1889820 T^{4} - 1128357424 T^{6} + 476406585350 T^{8} - 1128357424 p^{4} T^{10} + 1889820 p^{8} T^{12} - 2000 p^{12} T^{14} + p^{16} T^{16}
23 1−40T+800T2−18248T3−19516T4+7340008T5−111492768T6+1524968904T7−5167201274T8+1524968904p2T9−111492768p4T10+7340008p6T11−19516p8T12−18248p10T13+800p12T14−40p14T15+p16T16 1 - 40 T + 800 T^{2} - 18248 T^{3} - 19516 T^{4} + 7340008 T^{5} - 111492768 T^{6} + 1524968904 T^{7} - 5167201274 T^{8} + 1524968904 p^{2} T^{9} - 111492768 p^{4} T^{10} + 7340008 p^{6} T^{11} - 19516 p^{8} T^{12} - 18248 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16}
29 1−2228T2+1857576T4−965861788T6+647582808974T8−965861788p4T10+1857576p8T12−2228p12T14+p16T16 1 - 2228 T^{2} + 1857576 T^{4} - 965861788 T^{6} + 647582808974 T^{8} - 965861788 p^{4} T^{10} + 1857576 p^{8} T^{12} - 2228 p^{12} T^{14} + p^{16} T^{16}
31 (1+48T+1740T2−1968T3−318442T4−1968p2T5+1740p4T6+48p6T7+p8T8)2 ( 1 + 48 T + 1740 T^{2} - 1968 T^{3} - 318442 T^{4} - 1968 p^{2} T^{5} + 1740 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2}
37 1−60T+1800T2+32652T3−6760976T4+249442548T5−2263719528T6−292468480836T7+19178941117086T8−292468480836p2T9−2263719528p4T10+249442548p6T11−6760976p8T12+32652p10T13+1800p12T14−60p14T15+p16T16 1 - 60 T + 1800 T^{2} + 32652 T^{3} - 6760976 T^{4} + 249442548 T^{5} - 2263719528 T^{6} - 292468480836 T^{7} + 19178941117086 T^{8} - 292468480836 p^{2} T^{9} - 2263719528 p^{4} T^{10} + 249442548 p^{6} T^{11} - 6760976 p^{8} T^{12} + 32652 p^{10} T^{13} + 1800 p^{12} T^{14} - 60 p^{14} T^{15} + p^{16} T^{16}
41 (1+76T+5776T2+267748T3+13703518T4+267748p2T5+5776p4T6+76p6T7+p8T8)2 ( 1 + 76 T + 5776 T^{2} + 267748 T^{3} + 13703518 T^{4} + 267748 p^{2} T^{5} + 5776 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2}
43 1+88T+3872T2+100696T3+1714468T4+169137736T5+13315542880T6+642392563656T7+29742475153158T8+642392563656p2T9+13315542880p4T10+169137736p6T11+1714468p8T12+100696p10T13+3872p12T14+88p14T15+p16T16 1 + 88 T + 3872 T^{2} + 100696 T^{3} + 1714468 T^{4} + 169137736 T^{5} + 13315542880 T^{6} + 642392563656 T^{7} + 29742475153158 T^{8} + 642392563656 p^{2} T^{9} + 13315542880 p^{4} T^{10} + 169137736 p^{6} T^{11} + 1714468 p^{8} T^{12} + 100696 p^{10} T^{13} + 3872 p^{12} T^{14} + 88 p^{14} T^{15} + p^{16} T^{16}
47 1−16T+128T2+128816T3+2412964T4−504683248T5+16062853504T6−177987325872T7−50194759450554T8−177987325872p2T9+16062853504p4T10−504683248p6T11+2412964p8T12+128816p10T13+128p12T14−16p14T15+p16T16 1 - 16 T + 128 T^{2} + 128816 T^{3} + 2412964 T^{4} - 504683248 T^{5} + 16062853504 T^{6} - 177987325872 T^{7} - 50194759450554 T^{8} - 177987325872 p^{2} T^{9} + 16062853504 p^{4} T^{10} - 504683248 p^{6} T^{11} + 2412964 p^{8} T^{12} + 128816 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16}
53 1+108T+5832T2+412884T3+23090224T4+11383740pT5+15734940120T6+223612868052T7−24915462238434T8+223612868052p2T9+15734940120p4T10+11383740p7T11+23090224p8T12+412884p10T13+5832p12T14+108p14T15+p16T16 1 + 108 T + 5832 T^{2} + 412884 T^{3} + 23090224 T^{4} + 11383740 p T^{5} + 15734940120 T^{6} + 223612868052 T^{7} - 24915462238434 T^{8} + 223612868052 p^{2} T^{9} + 15734940120 p^{4} T^{10} + 11383740 p^{7} T^{11} + 23090224 p^{8} T^{12} + 412884 p^{10} T^{13} + 5832 p^{12} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16}
59 1−25500T2+290547640T4−1939529752884T6+8319211197765870T8−1939529752884p4T10+290547640p8T12−25500p12T14+p16T16 1 - 25500 T^{2} + 290547640 T^{4} - 1939529752884 T^{6} + 8319211197765870 T^{8} - 1939529752884 p^{4} T^{10} + 290547640 p^{8} T^{12} - 25500 p^{12} T^{14} + p^{16} T^{16}
61 (1+132T+13872T2+941868T3+61614014T4+941868p2T5+13872p4T6+132p6T7+p8T8)2 ( 1 + 132 T + 13872 T^{2} + 941868 T^{3} + 61614014 T^{4} + 941868 p^{2} T^{5} + 13872 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} )^{2}
67 1+216T+23328T2+2220888T3+213731236T4+17658282696T5+1294438543200T6+97283636525256T7+7010705853400710T8+97283636525256p2T9+1294438543200p4T10+17658282696p6T11+213731236p8T12+2220888p10T13+23328p12T14+216p14T15+p16T16 1 + 216 T + 23328 T^{2} + 2220888 T^{3} + 213731236 T^{4} + 17658282696 T^{5} + 1294438543200 T^{6} + 97283636525256 T^{7} + 7010705853400710 T^{8} + 97283636525256 p^{2} T^{9} + 1294438543200 p^{4} T^{10} + 17658282696 p^{6} T^{11} + 213731236 p^{8} T^{12} + 2220888 p^{10} T^{13} + 23328 p^{12} T^{14} + 216 p^{14} T^{15} + p^{16} T^{16}
71 (1−120T+20340T2−1543656T3+149872550T4−1543656p2T5+20340p4T6−120p6T7+p8T8)2 ( 1 - 120 T + 20340 T^{2} - 1543656 T^{3} + 149872550 T^{4} - 1543656 p^{2} T^{5} + 20340 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2}
73 1+208T+21632T2+2121136T3+133201276T4+248473360T5−580118578304T6−88473819716496T7−8873634681591930T8−88473819716496p2T9−580118578304p4T10+248473360p6T11+133201276p8T12+2121136p10T13+21632p12T14+208p14T15+p16T16 1 + 208 T + 21632 T^{2} + 2121136 T^{3} + 133201276 T^{4} + 248473360 T^{5} - 580118578304 T^{6} - 88473819716496 T^{7} - 8873634681591930 T^{8} - 88473819716496 p^{2} T^{9} - 580118578304 p^{4} T^{10} + 248473360 p^{6} T^{11} + 133201276 p^{8} T^{12} + 2121136 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 p^{14} T^{15} + p^{16} T^{16}
79 1−32504T2+462697980T4−4035012008776T6+27090604999745414T8−4035012008776p4T10+462697980p8T12−32504p12T14+p16T16 1 - 32504 T^{2} + 462697980 T^{4} - 4035012008776 T^{6} + 27090604999745414 T^{8} - 4035012008776 p^{4} T^{10} + 462697980 p^{8} T^{12} - 32504 p^{12} T^{14} + p^{16} T^{16}
83 1−336T+56448T2−7845648T3+1037744740T4−116459238960T5+11328785476992T6−1064762407254768T7+93992795964479238T8−1064762407254768p2T9+11328785476992p4T10−116459238960p6T11+1037744740p8T12−7845648p10T13+56448p12T14−336p14T15+p16T16 1 - 336 T + 56448 T^{2} - 7845648 T^{3} + 1037744740 T^{4} - 116459238960 T^{5} + 11328785476992 T^{6} - 1064762407254768 T^{7} + 93992795964479238 T^{8} - 1064762407254768 p^{2} T^{9} + 11328785476992 p^{4} T^{10} - 116459238960 p^{6} T^{11} + 1037744740 p^{8} T^{12} - 7845648 p^{10} T^{13} + 56448 p^{12} T^{14} - 336 p^{14} T^{15} + p^{16} T^{16}
89 1−25856T2+291349884T4−1811616834304T6+9967156810684550T8−1811616834304p4T10+291349884p8T12−25856p12T14+p16T16 1 - 25856 T^{2} + 291349884 T^{4} - 1811616834304 T^{6} + 9967156810684550 T^{8} - 1811616834304 p^{4} T^{10} + 291349884 p^{8} T^{12} - 25856 p^{12} T^{14} + p^{16} T^{16}
97 1+208T+21632T2+956336T3−203065348T4−29702194672T5−1328057611392T6+139759626438000T7+32182715530229254T8+139759626438000p2T9−1328057611392p4T10−29702194672p6T11−203065348p8T12+956336p10T13+21632p12T14+208p14T15+p16T16 1 + 208 T + 21632 T^{2} + 956336 T^{3} - 203065348 T^{4} - 29702194672 T^{5} - 1328057611392 T^{6} + 139759626438000 T^{7} + 32182715530229254 T^{8} + 139759626438000 p^{2} T^{9} - 1328057611392 p^{4} T^{10} - 29702194672 p^{6} T^{11} - 203065348 p^{8} T^{12} + 956336 p^{10} T^{13} + 21632 p^{12} T^{14} + 208 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

−4.05371527180654207206207355446, −3.76838134965731607558421645743, −3.73660696040616318198794468104, −3.66245239908368265561829071525, −3.45745919215482034589927655656, −3.32007384724287212576809728490, −3.30317241227808760400548906744, −3.25414433350991750098136237606, −3.03659984838017652073938272945, −3.01152903926381871594545143044, −2.79852182898296527583185591415, −2.44225971568453772716518243880, −2.24974049631523435540852144840, −2.15335987779485778129413460005, −2.05854793175559144959742431310, −1.86958297945773156469470601551, −1.55507737203613120679764629554, −1.46672640223742759371929411765, −1.44678456012447289212102404805, −1.30813122649260456247396024048, −1.22542057587887873392484207828, −1.14446651728387048762505312516, −0.57483693624640599727664606790, −0.31854129159442069006189263696, −0.23577505461654047839190735307, 0.23577505461654047839190735307, 0.31854129159442069006189263696, 0.57483693624640599727664606790, 1.14446651728387048762505312516, 1.22542057587887873392484207828, 1.30813122649260456247396024048, 1.44678456012447289212102404805, 1.46672640223742759371929411765, 1.55507737203613120679764629554, 1.86958297945773156469470601551, 2.05854793175559144959742431310, 2.15335987779485778129413460005, 2.24974049631523435540852144840, 2.44225971568453772716518243880, 2.79852182898296527583185591415, 3.01152903926381871594545143044, 3.03659984838017652073938272945, 3.25414433350991750098136237606, 3.30317241227808760400548906744, 3.32007384724287212576809728490, 3.45745919215482034589927655656, 3.66245239908368265561829071525, 3.73660696040616318198794468104, 3.76838134965731607558421645743, 4.05371527180654207206207355446

Graph of the ZZ-function along the critical line

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