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Properties

Label 16-31e16-1.1-c1e8-0-2
Degree 1616
Conductor 7.274×10237.274\times 10^{23}
Sign 11
Analytic cond. 1.20227×1071.20227\times 10^{7}
Root an. cond. 2.770132.77013
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  − 6·2-s + 6·3-s + 23·4-s − 4·5-s − 36·6-s − 7·7-s − 72·8-s + 13·9-s + 24·10-s − 2·11-s + 138·12-s + 6·13-s + 42·14-s − 24·15-s + 199·16-s − 8·17-s − 78·18-s − 5·19-s − 92·20-s − 42·21-s + 12·22-s − 32·23-s − 432·24-s + 26·25-s − 36·26-s − 2·27-s − 161·28-s + ⋯
L(s)  = 1  − 4.24·2-s + 3.46·3-s + 23/2·4-s − 1.78·5-s − 14.6·6-s − 2.64·7-s − 25.4·8-s + 13/3·9-s + 7.58·10-s − 0.603·11-s + 39.8·12-s + 1.66·13-s + 11.2·14-s − 6.19·15-s + 49.7·16-s − 1.94·17-s − 18.3·18-s − 1.14·19-s − 20.5·20-s − 9.16·21-s + 2.55·22-s − 6.67·23-s − 88.1·24-s + 26/5·25-s − 7.06·26-s − 0.384·27-s − 30.4·28-s + ⋯

Functional equation

Λ(s)=((3116)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((3116)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 311631^{16}
Sign: 11
Analytic conductor: 1.20227×1071.20227\times 10^{7}
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 3116, ( :[1/2]8), 1)(16,\ 31^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.048839436050.04883943605
L(12)L(\frac12) ≈\approx 0.048839436050.04883943605
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)
bad31 1 1
good2 (1+3T+pT2+T4+p3T6+3p3T7+p4T8)2 ( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
3 1−2pT+23T2−58T3+4p3T4−148T5+47pT6−94T7+43T8−94pT9+47p3T10−148p3T11+4p7T12−58p5T13+23p6T14−2p8T15+p8T16 1 - 2 p T + 23 T^{2} - 58 T^{3} + 4 p^{3} T^{4} - 148 T^{5} + 47 p T^{6} - 94 T^{7} + 43 T^{8} - 94 p T^{9} + 47 p^{3} T^{10} - 148 p^{3} T^{11} + 4 p^{7} T^{12} - 58 p^{5} T^{13} + 23 p^{6} T^{14} - 2 p^{8} T^{15} + p^{8} T^{16}
5 (1+T−4T2+pT3+p2T4)4 ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{4}
7 1+pT+37T2+134T3+458T4+1471T5+4694T6+14192T7+39103T8+14192pT9+4694p2T10+1471p3T11+458p4T12+134p5T13+37p6T14+p8T15+p8T16 1 + p T + 37 T^{2} + 134 T^{3} + 458 T^{4} + 1471 T^{5} + 4694 T^{6} + 14192 T^{7} + 39103 T^{8} + 14192 p T^{9} + 4694 p^{2} T^{10} + 1471 p^{3} T^{11} + 458 p^{4} T^{12} + 134 p^{5} T^{13} + 37 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16}
11 1+2T+pT2+58T3+116T4−164T5+39pT6−4756T7−24153T8−4756pT9+39p3T10−164p3T11+116p4T12+58p5T13+p7T14+2p7T15+p8T16 1 + 2 T + p T^{2} + 58 T^{3} + 116 T^{4} - 164 T^{5} + 39 p T^{6} - 4756 T^{7} - 24153 T^{8} - 4756 p T^{9} + 39 p^{3} T^{10} - 164 p^{3} T^{11} + 116 p^{4} T^{12} + 58 p^{5} T^{13} + p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
13 1−6T+33T2−38T3−12T4+1132T5+127pT6−16194T7+133363T8−16194pT9+127p3T10+1132p3T11−12p4T12−38p5T13+33p6T14−6p7T15+p8T16 1 - 6 T + 33 T^{2} - 38 T^{3} - 12 T^{4} + 1132 T^{5} + 127 p T^{6} - 16194 T^{7} + 133363 T^{8} - 16194 p T^{9} + 127 p^{3} T^{10} + 1132 p^{3} T^{11} - 12 p^{4} T^{12} - 38 p^{5} T^{13} + 33 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
17 1+8T+57T2+276T3+1508T4+6624T5+35159T6+145358T7+660723T8+145358pT9+35159p2T10+6624p3T11+1508p4T12+276p5T13+57p6T14+8p7T15+p8T16 1 + 8 T + 57 T^{2} + 276 T^{3} + 1508 T^{4} + 6624 T^{5} + 35159 T^{6} + 145358 T^{7} + 660723 T^{8} + 145358 p T^{9} + 35159 p^{2} T^{10} + 6624 p^{3} T^{11} + 1508 p^{4} T^{12} + 276 p^{5} T^{13} + 57 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
19 1+5T+29T2−70T3−590T4−4435T5−2714T6+42900T7+426339T8+42900pT9−2714p2T10−4435p3T11−590p4T12−70p5T13+29p6T14+5p7T15+p8T16 1 + 5 T + 29 T^{2} - 70 T^{3} - 590 T^{4} - 4435 T^{5} - 2714 T^{6} + 42900 T^{7} + 426339 T^{8} + 42900 p T^{9} - 2714 p^{2} T^{10} - 4435 p^{3} T^{11} - 590 p^{4} T^{12} - 70 p^{5} T^{13} + 29 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
23 (1+16T+113T2+570T3+2741T4+570pT5+113p2T6+16p3T7+p4T8)2 ( 1 + 16 T + 113 T^{2} + 570 T^{3} + 2741 T^{4} + 570 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1+11T2+90T3+661T4+90pT5+11p2T6+p4T8)2 ( 1 + 11 T^{2} + 90 T^{3} + 661 T^{4} + 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2}
37 (1+2T−33T2+2pT3+p2T4)4 ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4}
41 1−7T+pT2−518T3+3626T4−15911T5+3954pT6−1177414T7+5416137T8−1177414pT9+3954p3T10−15911p3T11+3626p4T12−518p5T13+p7T14−7p7T15+p8T16 1 - 7 T + p T^{2} - 518 T^{3} + 3626 T^{4} - 15911 T^{5} + 3954 p T^{6} - 1177414 T^{7} + 5416137 T^{8} - 1177414 p T^{9} + 3954 p^{3} T^{10} - 15911 p^{3} T^{11} + 3626 p^{4} T^{12} - 518 p^{5} T^{13} + p^{7} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16}
43 1+4T+pT2−328T3−2092T4−17288T5−42379T6+635666T7+2926363T8+635666pT9−42379p2T10−17288p3T11−2092p4T12−328p5T13+p7T14+4p7T15+p8T16 1 + 4 T + p T^{2} - 328 T^{3} - 2092 T^{4} - 17288 T^{5} - 42379 T^{6} + 635666 T^{7} + 2926363 T^{8} + 635666 p T^{9} - 42379 p^{2} T^{10} - 17288 p^{3} T^{11} - 2092 p^{4} T^{12} - 328 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
47 (1+8T+17T2+380T3+4721T4+380pT5+17p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1+4T−27T2−588T3−4432T4−1608T5+100691T6+1155796T7+6693663T8+1155796pT9+100691p2T10−1608p3T11−4432p4T12−588p5T13−27p6T14+4p7T15+p8T16 1 + 4 T - 27 T^{2} - 588 T^{3} - 4432 T^{4} - 1608 T^{5} + 100691 T^{6} + 1155796 T^{7} + 6693663 T^{8} + 1155796 p T^{9} + 100691 p^{2} T^{10} - 1608 p^{3} T^{11} - 4432 p^{4} T^{12} - 588 p^{5} T^{13} - 27 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
59 1−5T+69T2+270T3−1990T4+37635T5−145834T6−485900T7+5823099T8−485900pT9−145834p2T10+37635p3T11−1990p4T12+270p5T13+69p6T14−5p7T15+p8T16 1 - 5 T + 69 T^{2} + 270 T^{3} - 1990 T^{4} + 37635 T^{5} - 145834 T^{6} - 485900 T^{7} + 5823099 T^{8} - 485900 p T^{9} - 145834 p^{2} T^{10} + 37635 p^{3} T^{11} - 1990 p^{4} T^{12} + 270 p^{5} T^{13} + 69 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
61 (1−6T+6T2−6pT3+p2T4)4 ( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4}
67 (1+8T−3T2+8pT3+p2T4)4 ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4}
71 1−27T+421T2−3318T3+6786T4+165549T5−1786666T6+6219816T7−615913T8+6219816pT9−1786666p2T10+165549p3T11+6786p4T12−3318p5T13+421p6T14−27p7T15+p8T16 1 - 27 T + 421 T^{2} - 3318 T^{3} + 6786 T^{4} + 165549 T^{5} - 1786666 T^{6} + 6219816 T^{7} - 615913 T^{8} + 6219816 p T^{9} - 1786666 p^{2} T^{10} + 165549 p^{3} T^{11} + 6786 p^{4} T^{12} - 3318 p^{5} T^{13} + 421 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16}
73 1−6T+33T2+1262T3−12972T4+61052T5+133831T6−7999944T7+53220103T8−7999944pT9+133831p2T10+61052p3T11−12972p4T12+1262p5T13+33p6T14−6p7T15+p8T16 1 - 6 T + 33 T^{2} + 1262 T^{3} - 12972 T^{4} + 61052 T^{5} + 133831 T^{6} - 7999944 T^{7} + 53220103 T^{8} - 7999944 p T^{9} + 133831 p^{2} T^{10} + 61052 p^{3} T^{11} - 12972 p^{4} T^{12} + 1262 p^{5} T^{13} + 33 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
79 1+10T+19T2−1570T3−21460T4−132020T5+181241T6+13939970T7+166725259T8+13939970pT9+181241p2T10−132020p3T11−21460p4T12−1570p5T13+19p6T14+10p7T15+p8T16 1 + 10 T + 19 T^{2} - 1570 T^{3} - 21460 T^{4} - 132020 T^{5} + 181241 T^{6} + 13939970 T^{7} + 166725259 T^{8} + 13939970 p T^{9} + 181241 p^{2} T^{10} - 132020 p^{3} T^{11} - 21460 p^{4} T^{12} - 1570 p^{5} T^{13} + 19 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16}
83 1−26T+483T2−5778T3+60548T4−592908T5+6213461T6−66709124T7+637073703T8−66709124pT9+6213461p2T10−592908p3T11+60548p4T12−5778p5T13+483p6T14−26p7T15+p8T16 1 - 26 T + 483 T^{2} - 5778 T^{3} + 60548 T^{4} - 592908 T^{5} + 6213461 T^{6} - 66709124 T^{7} + 637073703 T^{8} - 66709124 p T^{9} + 6213461 p^{2} T^{10} - 592908 p^{3} T^{11} + 60548 p^{4} T^{12} - 5778 p^{5} T^{13} + 483 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16}
89 (1+10T+71T2+1290T3+19001T4+1290pT5+71p2T6+10p3T7+p4T8)2 ( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 1290 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 (1+13T+192T2+2195T3+31031T4+2195pT5+192p2T6+13p3T7+p4T8)2 ( 1 + 13 T + 192 T^{2} + 2195 T^{3} + 31031 T^{4} + 2195 p T^{5} + 192 p^{2} T^{6} + 13 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

−4.19181902599993387236173495526, −3.82074414892482064786618458628, −3.75667651272346123979627738810, −3.66133705192535536103874893740, −3.56023356987468585740590288174, −3.42648652328988362792831464213, −3.31937935843777864215981122886, −3.31145309853263925104729576062, −3.05879889494102784883176069138, −3.03548249419434629675523859822, −2.95220889952915175334062271893, −2.69264545887661440283173261105, −2.63269514415010715267135077926, −2.29607635489839945317745321525, −2.28199765459870036274052816691, −2.14590047692293573887586464612, −1.97254207134369230581726673355, −1.92117219814172768981099265243, −1.91424092126641772955543096281, −1.47557155684217227106171692713, −1.20011809796997359784596369741, −1.02874398411419716296613537054, −0.38270290903840431955197540654, −0.31011149207910842362027125560, −0.14252684859426603654556820341, 0.14252684859426603654556820341, 0.31011149207910842362027125560, 0.38270290903840431955197540654, 1.02874398411419716296613537054, 1.20011809796997359784596369741, 1.47557155684217227106171692713, 1.91424092126641772955543096281, 1.92117219814172768981099265243, 1.97254207134369230581726673355, 2.14590047692293573887586464612, 2.28199765459870036274052816691, 2.29607635489839945317745321525, 2.63269514415010715267135077926, 2.69264545887661440283173261105, 2.95220889952915175334062271893, 3.03548249419434629675523859822, 3.05879889494102784883176069138, 3.31145309853263925104729576062, 3.31937935843777864215981122886, 3.42648652328988362792831464213, 3.56023356987468585740590288174, 3.66133705192535536103874893740, 3.75667651272346123979627738810, 3.82074414892482064786618458628, 4.19181902599993387236173495526

Graph of the ZZ-function along the critical line

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