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Properties

Label 16-31e16-1.1-c1e8-0-12
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  + 4·2-s − 4·3-s + 8·4-s − 4·5-s − 16·6-s + 3·7-s + 18·8-s + 13·9-s − 16·10-s − 2·11-s − 32·12-s − 4·13-s + 12·14-s + 16·15-s + 34·16-s + 2·17-s + 52·18-s + 5·19-s − 32·20-s − 12·21-s − 8·22-s + 28·23-s − 72·24-s + 26·25-s − 16·26-s − 32·27-s + 24·28-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 4·4-s − 1.78·5-s − 6.53·6-s + 1.13·7-s + 6.36·8-s + 13/3·9-s − 5.05·10-s − 0.603·11-s − 9.23·12-s − 1.10·13-s + 3.20·14-s + 4.13·15-s + 17/2·16-s + 0.485·17-s + 12.2·18-s + 1.14·19-s − 7.15·20-s − 2.61·21-s − 1.70·22-s + 5.83·23-s − 14.6·24-s + 26/5·25-s − 3.13·26-s − 6.15·27-s + 4.53·28-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 46.9346980546.93469805
L(12)L(\frac12) ≈\approx 46.9346980546.93469805
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−pT+pT2−5T3+11T4−5pT5+p3T6−p4T7+p4T8)2 ( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2}
3 1+4T+pT2−8T3−4pT4−8T5−19T6+22pT7+283T8+22p2T9−19p2T10−8p3T11−4p5T12−8p5T13+p7T14+4p7T15+p8T16 1 + 4 T + p T^{2} - 8 T^{3} - 4 p T^{4} - 8 T^{5} - 19 T^{6} + 22 p T^{7} + 283 T^{8} + 22 p^{2} T^{9} - 19 p^{2} T^{10} - 8 p^{3} T^{11} - 4 p^{5} T^{12} - 8 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} 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 1−3T−3T2+34T3−102T4+181T5−26T6−264pT7+8023T8−264p2T9−26p2T10+181p3T11−102p4T12+34p5T13−3p6T14−3p7T15+p8T16 1 - 3 T - 3 T^{2} + 34 T^{3} - 102 T^{4} + 181 T^{5} - 26 T^{6} - 264 p T^{7} + 8023 T^{8} - 264 p^{2} T^{9} - 26 p^{2} T^{10} + 181 p^{3} T^{11} - 102 p^{4} T^{12} + 34 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} 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+4T+pT2−88T3−532T4−1928T5−109T6+23966T7+124003T8+23966pT9−109p2T10−1928p3T11−532p4T12−88p5T13+p7T14+4p7T15+p8T16 1 + 4 T + p T^{2} - 88 T^{3} - 532 T^{4} - 1928 T^{5} - 109 T^{6} + 23966 T^{7} + 124003 T^{8} + 23966 p T^{9} - 109 p^{2} T^{10} - 1928 p^{3} T^{11} - 532 p^{4} T^{12} - 88 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
17 1−2T−3T2+126T3−532T4+84T5+3479T6−28082T7+52803T8−28082pT9+3479p2T10+84p3T11−532p4T12+126p5T13−3p6T14−2p7T15+p8T16 1 - 2 T - 3 T^{2} + 126 T^{3} - 532 T^{4} + 84 T^{5} + 3479 T^{6} - 28082 T^{7} + 52803 T^{8} - 28082 p T^{9} + 3479 p^{2} T^{10} + 84 p^{3} T^{11} - 532 p^{4} T^{12} + 126 p^{5} T^{13} - 3 p^{6} T^{14} - 2 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 (1−14T+113T2−750T3+4121T4−750pT5+113p2T6−14p3T7+p4T8)2 ( 1 - 14 T + 113 T^{2} - 750 T^{3} + 4121 T^{4} - 750 p T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1−10T+11T2+10pT3−2239T4+10p2T5+11p2T6−10p3T7+p4T8)2 ( 1 - 10 T + 11 T^{2} + 10 p T^{3} - 2239 T^{4} + 10 p^{2} T^{5} + 11 p^{2} T^{6} - 10 p^{3} T^{7} + 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 1−6T+63T2+22T3−372T4+19372T5−833pT6+91506T7+3060523T8+91506pT9−833p3T10+19372p3T11−372p4T12+22p5T13+63p6T14−6p7T15+p8T16 1 - 6 T + 63 T^{2} + 22 T^{3} - 372 T^{4} + 19372 T^{5} - 833 p T^{6} + 91506 T^{7} + 3060523 T^{8} + 91506 p T^{9} - 833 p^{3} T^{10} + 19372 p^{3} T^{11} - 372 p^{4} T^{12} + 22 p^{5} T^{13} + 63 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
47 (1−12T+17T2+60T3+961T4+60pT5+17p2T6−12p3T7+p4T8)2 ( 1 - 12 T + 17 T^{2} + 60 T^{3} + 961 T^{4} + 60 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1−16T+213T2−2088T3+21008T4−187488T5+1667411T6−13076524T7+100516383T8−13076524pT9+1667411p2T10−187488p3T11+21008p4T12−2088p5T13+213p6T14−16p7T15+p8T16 1 - 16 T + 213 T^{2} - 2088 T^{3} + 21008 T^{4} - 187488 T^{5} + 1667411 T^{6} - 13076524 T^{7} + 100516383 T^{8} - 13076524 p T^{9} + 1667411 p^{2} T^{10} - 187488 p^{3} T^{11} + 21008 p^{4} T^{12} - 2088 p^{5} T^{13} + 213 p^{6} T^{14} - 16 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+23T+221T2−18T3−21614T4−255201T5−803866T6+12228416T7+191155087T8+12228416pT9−803866p2T10−255201p3T11−21614p4T12−18p5T13+221p6T14+23p7T15+p8T16 1 + 23 T + 221 T^{2} - 18 T^{3} - 21614 T^{4} - 255201 T^{5} - 803866 T^{6} + 12228416 T^{7} + 191155087 T^{8} + 12228416 p T^{9} - 803866 p^{2} T^{10} - 255201 p^{3} T^{11} - 21614 p^{4} T^{12} - 18 p^{5} T^{13} + 221 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16}
73 1+14T+193T2+1162T3+10388T4+28532T5+925511T6+7337176T7+107602183T8+7337176pT9+925511p2T10+28532p3T11+10388p4T12+1162p5T13+193p6T14+14p7T15+p8T16 1 + 14 T + 193 T^{2} + 1162 T^{3} + 10388 T^{4} + 28532 T^{5} + 925511 T^{6} + 7337176 T^{7} + 107602183 T^{8} + 7337176 p T^{9} + 925511 p^{2} T^{10} + 28532 p^{3} T^{11} + 10388 p^{4} T^{12} + 1162 p^{5} T^{13} + 193 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
79 1−20T+319T2−2800T3+25940T4−208760T5+2905241T6−31993630T7+344475259T8−31993630pT9+2905241p2T10−208760p3T11+25940p4T12−2800p5T13+319p6T14−20p7T15+p8T16 1 - 20 T + 319 T^{2} - 2800 T^{3} + 25940 T^{4} - 208760 T^{5} + 2905241 T^{6} - 31993630 T^{7} + 344475259 T^{8} - 31993630 p T^{9} + 2905241 p^{2} T^{10} - 208760 p^{3} T^{11} + 25940 p^{4} T^{12} - 2800 p^{5} T^{13} + 319 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16}
83 1+14T+3T2−1578T3−19132T4−149148T5−347179T6+14598716T7+238036263T8+14598716pT9−347179p2T10−149148p3T11−19132p4T12−1578p5T13+3p6T14+14p7T15+p8T16 1 + 14 T + 3 T^{2} - 1578 T^{3} - 19132 T^{4} - 149148 T^{5} - 347179 T^{6} + 14598716 T^{7} + 238036263 T^{8} + 14598716 p T^{9} - 347179 p^{2} T^{10} - 149148 p^{3} T^{11} - 19132 p^{4} T^{12} - 1578 p^{5} T^{13} + 3 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
89 (1−20T+71T2+690T3−7699T4+690pT5+71p2T6−20p3T7+p4T8)2 ( 1 - 20 T + 71 T^{2} + 690 T^{3} - 7699 T^{4} + 690 p T^{5} + 71 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 (1−27T+192T2+955T3−23289T4+955pT5+192p2T6−27p3T7+p4T8)2 ( 1 - 27 T + 192 T^{2} + 955 T^{3} - 23289 T^{4} + 955 p T^{5} + 192 p^{2} T^{6} - 27 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

−4.52435980719603680001100598172, −4.39448573548930614357429059339, −4.29061794898971746077756115836, −3.83449799193111635304736941703, −3.81442691856251828230523749240, −3.80676860411639364076520757657, −3.74461803141583286952812632724, −3.67352087325942085493796337293, −3.36611556884902977325671845325, −3.10556657297953477120340437343, −2.91391289803254885743235969231, −2.80098343118264488974724986299, −2.73699340012843923509883138040, −2.52205750128830774953872616689, −2.49922807003681699770312221161, −2.35380088071299758194791094981, −2.26903554387706356422805868926, −1.54690774430578196714892649507, −1.50705092370514977211226580736, −1.46552546852412245497800009767, −1.07486563567061832500900629791, −1.04939137955881516980091942530, −0.944001138226368537933677893660, −0.73049204778867155993852864620, −0.56906579653739646137767397002, 0.56906579653739646137767397002, 0.73049204778867155993852864620, 0.944001138226368537933677893660, 1.04939137955881516980091942530, 1.07486563567061832500900629791, 1.46552546852412245497800009767, 1.50705092370514977211226580736, 1.54690774430578196714892649507, 2.26903554387706356422805868926, 2.35380088071299758194791094981, 2.49922807003681699770312221161, 2.52205750128830774953872616689, 2.73699340012843923509883138040, 2.80098343118264488974724986299, 2.91391289803254885743235969231, 3.10556657297953477120340437343, 3.36611556884902977325671845325, 3.67352087325942085493796337293, 3.74461803141583286952812632724, 3.80676860411639364076520757657, 3.81442691856251828230523749240, 3.83449799193111635304736941703, 4.29061794898971746077756115836, 4.39448573548930614357429059339, 4.52435980719603680001100598172

Graph of the ZZ-function along the critical line

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