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Properties

Label 16-31e16-1.1-c1e8-0-10
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 − 3-s + 23·4-s + 6·5-s + 6·6-s + 3·7-s − 72·8-s − 2·9-s − 36·10-s + 2·11-s − 23·12-s − 6·13-s − 18·14-s − 6·15-s + 199·16-s + 3·17-s + 12·18-s + 5·19-s + 138·20-s − 3·21-s − 12·22-s + 22·23-s + 72·24-s + 31·25-s + 36·26-s − 3·27-s + 69·28-s + ⋯
L(s)  = 1  − 4.24·2-s − 0.577·3-s + 23/2·4-s + 2.68·5-s + 2.44·6-s + 1.13·7-s − 25.4·8-s − 2/3·9-s − 11.3·10-s + 0.603·11-s − 6.63·12-s − 1.66·13-s − 4.81·14-s − 1.54·15-s + 49.7·16-s + 0.727·17-s + 2.82·18-s + 1.14·19-s + 30.8·20-s − 0.654·21-s − 2.55·22-s + 4.58·23-s + 14.6·24-s + 31/5·25-s + 7.06·26-s − 0.577·27-s + 13.0·28-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 3.3689305953.368930595
L(12)L(\frac12) ≈\approx 3.3689305953.368930595
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+T+pT2+8T3+8T4−7T5+2pT6−56T7−137T8−56pT9+2p3T10−7p3T11+8p4T12+8p5T13+p7T14+p7T15+p8T16 1 + T + p T^{2} + 8 T^{3} + 8 T^{4} - 7 T^{5} + 2 p T^{6} - 56 T^{7} - 137 T^{8} - 56 p T^{9} + 2 p^{3} T^{10} - 7 p^{3} T^{11} + 8 p^{4} T^{12} + 8 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16}
5 (1−3T−2T2−3T3+51T4−3pT5−2p2T6−3p3T7+p4T8)2 ( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
7 1−3T+pT2−36T3+108T4−219T5+122pT6−2628T7+5483T8−2628pT9+122p3T10−219p3T11+108p4T12−36p5T13+p7T14−3p7T15+p8T16 1 - 3 T + p T^{2} - 36 T^{3} + 108 T^{4} - 219 T^{5} + 122 p T^{6} - 2628 T^{7} + 5483 T^{8} - 2628 p T^{9} + 122 p^{3} T^{10} - 219 p^{3} T^{11} + 108 p^{4} T^{12} - 36 p^{5} T^{13} + p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16}
11 1−2T−9T2+42T3−124T4+84T5+629T6−3734T7+13467T8−3734pT9+629p2T10+84p3T11−124p4T12+42p5T13−9p6T14−2p7T15+p8T16 1 - 2 T - 9 T^{2} + 42 T^{3} - 124 T^{4} + 84 T^{5} + 629 T^{6} - 3734 T^{7} + 13467 T^{8} - 3734 p T^{9} + 629 p^{2} T^{10} + 84 p^{3} T^{11} - 124 p^{4} T^{12} + 42 p^{5} T^{13} - 9 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
13 1+6T+pT2−102T3−792T4−2892T5−1009T6+36864T7+214223T8+36864pT9−1009p2T10−2892p3T11−792p4T12−102p5T13+p7T14+6p7T15+p8T16 1 + 6 T + p T^{2} - 102 T^{3} - 792 T^{4} - 2892 T^{5} - 1009 T^{6} + 36864 T^{7} + 214223 T^{8} + 36864 p T^{9} - 1009 p^{2} T^{10} - 2892 p^{3} T^{11} - 792 p^{4} T^{12} - 102 p^{5} T^{13} + p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
17 1−3T+7T2+144T3−732T4+1641T5+1934T6−49698T7+169253T8−49698pT9+1934p2T10+1641p3T11−732p4T12+144p5T13+7p6T14−3p7T15+p8T16 1 - 3 T + 7 T^{2} + 144 T^{3} - 732 T^{4} + 1641 T^{5} + 1934 T^{6} - 49698 T^{7} + 169253 T^{8} - 49698 p T^{9} + 1934 p^{2} T^{10} + 1641 p^{3} T^{11} - 732 p^{4} T^{12} + 144 p^{5} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16}
19 1−5T+pT2−160T3+800T4−2765T5+914pT6−88480T7+312079T8−88480pT9+914p3T10−2765p3T11+800p4T12−160p5T13+p7T14−5p7T15+p8T16 1 - 5 T + p T^{2} - 160 T^{3} + 800 T^{4} - 2765 T^{5} + 914 p T^{6} - 88480 T^{7} + 312079 T^{8} - 88480 p T^{9} + 914 p^{3} T^{10} - 2765 p^{3} T^{11} + 800 p^{4} T^{12} - 160 p^{5} T^{13} + p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
23 (1−11T+38T2−125T3+821T4−125pT5+38p2T6−11p3T7+p4T8)2 ( 1 - 11 T + 38 T^{2} - 125 T^{3} + 821 T^{4} - 125 p T^{5} + 38 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1−5T+31T2−115T3+96T4−115pT5+31p2T6−5p3T7+p4T8)2 ( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
37 (1−4T−57T2+4T3+3368T4+4pT5−57p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 57 T^{2} + 4 T^{3} + 3368 T^{4} + 4 p T^{5} - 57 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
41 1+8T+pT2−528T3−6224T4−39696T5−2521T6+1579436T7+15159727T8+1579436pT9−2521p2T10−39696p3T11−6224p4T12−528p5T13+p7T14+8p7T15+p8T16 1 + 8 T + p T^{2} - 528 T^{3} - 6224 T^{4} - 39696 T^{5} - 2521 T^{6} + 1579436 T^{7} + 15159727 T^{8} + 1579436 p T^{9} - 2521 p^{2} T^{10} - 39696 p^{3} T^{11} - 6224 p^{4} T^{12} - 528 p^{5} T^{13} + p^{7} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
43 1+T+28T2+443T3−232T4−6152T5−2224T6−797146T7−5551967T8−797146pT9−2224p2T10−6152p3T11−232p4T12+443p5T13+28p6T14+p7T15+p8T16 1 + T + 28 T^{2} + 443 T^{3} - 232 T^{4} - 6152 T^{5} - 2224 T^{6} - 797146 T^{7} - 5551967 T^{8} - 797146 p T^{9} - 2224 p^{2} T^{10} - 6152 p^{3} T^{11} - 232 p^{4} T^{12} + 443 p^{5} T^{13} + 28 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16}
47 (1−7T−23T2+385T3−1284T4+385pT5−23p2T6−7p3T7+p4T8)2 ( 1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 385 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1+21T+323T2+3588T3+35568T4+324633T5+2767366T6+22572954T7+167761313T8+22572954pT9+2767366p2T10+324633p3T11+35568p4T12+3588p5T13+323p6T14+21p7T15+p8T16 1 + 21 T + 323 T^{2} + 3588 T^{3} + 35568 T^{4} + 324633 T^{5} + 2767366 T^{6} + 22572954 T^{7} + 167761313 T^{8} + 22572954 p T^{9} + 2767366 p^{2} T^{10} + 324633 p^{3} T^{11} + 35568 p^{4} T^{12} + 3588 p^{5} T^{13} + 323 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16}
59 1+5T−T2−900T3−8060T4−17835T5+134126T6+3072320T7+18108079T8+3072320pT9+134126p2T10−17835p3T11−8060p4T12−900p5T13−p6T14+5p7T15+p8T16 1 + 5 T - T^{2} - 900 T^{3} - 8060 T^{4} - 17835 T^{5} + 134126 T^{6} + 3072320 T^{7} + 18108079 T^{8} + 3072320 p T^{9} + 134126 p^{2} T^{10} - 17835 p^{3} T^{11} - 8060 p^{4} T^{12} - 900 p^{5} T^{13} - p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
61 (1−4T+46T2−4pT3+p2T4)4 ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4}
67 (1−4T−117T2+4T3+12128T4+4pT5−117p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 117 T^{2} + 4 T^{3} + 12128 T^{4} + 4 p T^{5} - 117 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
71 1−7T−4T2+1227T3−13364T4+49044T5+214904T6−6815734T7+62113357T8−6815734pT9+214904p2T10+49044p3T11−13364p4T12+1227p5T13−4p6T14−7p7T15+p8T16 1 - 7 T - 4 T^{2} + 1227 T^{3} - 13364 T^{4} + 49044 T^{5} + 214904 T^{6} - 6815734 T^{7} + 62113357 T^{8} - 6815734 p T^{9} + 214904 p^{2} T^{10} + 49044 p^{3} T^{11} - 13364 p^{4} T^{12} + 1227 p^{5} T^{13} - 4 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16}
73 1+21T+208T2−297T3−27162T4−338082T5−1041544T6+15918264T7+260450783T8+15918264pT9−1041544p2T10−338082p3T11−27162p4T12−297p5T13+208p6T14+21p7T15+p8T16 1 + 21 T + 208 T^{2} - 297 T^{3} - 27162 T^{4} - 338082 T^{5} - 1041544 T^{6} + 15918264 T^{7} + 260450783 T^{8} + 15918264 p T^{9} - 1041544 p^{2} T^{10} - 338082 p^{3} T^{11} - 27162 p^{4} T^{12} - 297 p^{5} T^{13} + 208 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16}
79 1+pT2−p3T6−p4T8−p5T10+p7T14+p8T16 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16}
83 1−14T+183T2−42T3−7462T4+179508T5+16211T6−9573326T7+219347043T8−9573326pT9+16211p2T10+179508p3T11−7462p4T12−42p5T13+183p6T14−14p7T15+p8T16 1 - 14 T + 183 T^{2} - 42 T^{3} - 7462 T^{4} + 179508 T^{5} + 16211 T^{6} - 9573326 T^{7} + 219347043 T^{8} - 9573326 p T^{9} + 16211 p^{2} T^{10} + 179508 p^{3} T^{11} - 7462 p^{4} T^{12} - 42 p^{5} T^{13} + 183 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16}
89 (1−5T−29T2+785T3−624T4+785pT5−29p2T6−5p3T7+p4T8)2 ( 1 - 5 T - 29 T^{2} + 785 T^{3} - 624 T^{4} + 785 p T^{5} - 29 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 (1+3T+182T2−345T3+16591T4−345pT5+182p2T6+3p3T7+p4T8)2 ( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 345 p T^{5} + 182 p^{2} T^{6} + 3 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

−4.47045329453765318506814609737, −4.13750987003398233053581278421, −4.09782730757576199477479132867, −3.76159487124326704021399026823, −3.39177766940073419419721899966, −3.30815963737098415541682036502, −3.13743565435312915766343593389, −3.08583520114776822871969633819, −3.08305451208115529839210966350, −2.91417491809159066820847584683, −2.81214609046502376923362719476, −2.72562334396295455238466597356, −2.72329667748480717692769130805, −2.32322323920619696404217915511, −2.00147103659247028781458586740, −1.99563207359567482999859754532, −1.98482014245594317101545805412, −1.70711799803223450735004374021, −1.56426143540442415862744340013, −1.36707271579727279434767534580, −0.995672606876184435381753046402, −0.983601736695836751077992218048, −0.983392101654024871022079181097, −0.53757221368283849994529731969, −0.53685880572123514149093412181, 0.53685880572123514149093412181, 0.53757221368283849994529731969, 0.983392101654024871022079181097, 0.983601736695836751077992218048, 0.995672606876184435381753046402, 1.36707271579727279434767534580, 1.56426143540442415862744340013, 1.70711799803223450735004374021, 1.98482014245594317101545805412, 1.99563207359567482999859754532, 2.00147103659247028781458586740, 2.32322323920619696404217915511, 2.72329667748480717692769130805, 2.72562334396295455238466597356, 2.81214609046502376923362719476, 2.91417491809159066820847584683, 3.08305451208115529839210966350, 3.08583520114776822871969633819, 3.13743565435312915766343593389, 3.30815963737098415541682036502, 3.39177766940073419419721899966, 3.76159487124326704021399026823, 4.09782730757576199477479132867, 4.13750987003398233053581278421, 4.47045329453765318506814609737

Graph of the ZZ-function along the critical line

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