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Properties

Label 16-31e16-1.1-c1e8-0-11
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  + 2·2-s + 2·3-s + 3·4-s + 8·5-s + 4·6-s − 2·7-s + 4·8-s + 5·9-s + 16·10-s + 2·11-s + 6·12-s + 2·13-s − 4·14-s + 16·15-s + 4·16-s + 6·17-s + 10·18-s − 6·19-s + 24·20-s − 4·21-s + 4·22-s + 8·23-s + 8·24-s − 4·25-s + 4·26-s + 6·27-s − 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 3.57·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 5/3·9-s + 5.05·10-s + 0.603·11-s + 1.73·12-s + 0.554·13-s − 1.06·14-s + 4.13·15-s + 16-s + 1.45·17-s + 2.35·18-s − 1.37·19-s + 5.36·20-s − 0.872·21-s + 0.852·22-s + 1.66·23-s + 1.63·24-s − 4/5·25-s + 0.784·26-s + 1.15·27-s − 1.13·28-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 103.1414557103.1414557
L(12)L(\frac12) ≈\approx 103.1414557103.1414557
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+T2+T4−p4T5+25T6−3pT7−3T8−3p2T9+25p2T10−p7T11+p4T12+p6T14−p8T15+p8T16 1 - p T + T^{2} + T^{4} - p^{4} T^{5} + 25 T^{6} - 3 p T^{7} - 3 T^{8} - 3 p^{2} T^{9} + 25 p^{2} T^{10} - p^{7} T^{11} + p^{4} T^{12} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16}
3 1−2T−T2+2pT3−4T4−4p2T5+65T6+20T7−113T8+20pT9+65p2T10−4p5T11−4p4T12+2p6T13−p6T14−2p7T15+p8T16 1 - 2 T - T^{2} + 2 p T^{3} - 4 T^{4} - 4 p^{2} T^{5} + 65 T^{6} + 20 T^{7} - 113 T^{8} + 20 p T^{9} + 65 p^{2} T^{10} - 4 p^{5} T^{11} - 4 p^{4} T^{12} + 2 p^{6} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
5 (1−T+pT2)8 ( 1 - T + p T^{2} )^{8}
7 1+2T−9T2−30T3+36T4+156T5−135T6−324T7+1527T8−324pT9−135p2T10+156p3T11+36p4T12−30p5T13−9p6T14+2p7T15+p8T16 1 + 2 T - 9 T^{2} - 30 T^{3} + 36 T^{4} + 156 T^{5} - 135 T^{6} - 324 T^{7} + 1527 T^{8} - 324 p T^{9} - 135 p^{2} T^{10} + 156 p^{3} T^{11} + 36 p^{4} T^{12} - 30 p^{5} T^{13} - 9 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
11 1−2T−T2−10T3−52T4−820T5+2241T6+1972T7+9183T8+1972pT9+2241p2T10−820p3T11−52p4T12−10p5T13−p6T14−2p7T15+p8T16 1 - 2 T - T^{2} - 10 T^{3} - 52 T^{4} - 820 T^{5} + 2241 T^{6} + 1972 T^{7} + 9183 T^{8} + 1972 p T^{9} + 2241 p^{2} T^{10} - 820 p^{3} T^{11} - 52 p^{4} T^{12} - 10 p^{5} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
13 1−2T−15T2+42T3+84T4−996T5+1839T6+5964T7−35169T8+5964pT9+1839p2T10−996p3T11+84p4T12+42p5T13−15p6T14−2p7T15+p8T16 1 - 2 T - 15 T^{2} + 42 T^{3} + 84 T^{4} - 996 T^{5} + 1839 T^{6} + 5964 T^{7} - 35169 T^{8} + 5964 p T^{9} + 1839 p^{2} T^{10} - 996 p^{3} T^{11} + 84 p^{4} T^{12} + 42 p^{5} T^{13} - 15 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
17 1−6T+T2+6pT3−324T4−84pT5+9215T6−1740T7−72793T8−1740pT9+9215p2T10−84p4T11−324p4T12+6p6T13+p6T14−6p7T15+p8T16 1 - 6 T + T^{2} + 6 p T^{3} - 324 T^{4} - 84 p T^{5} + 9215 T^{6} - 1740 T^{7} - 72793 T^{8} - 1740 p T^{9} + 9215 p^{2} T^{10} - 84 p^{4} T^{11} - 324 p^{4} T^{12} + 6 p^{6} T^{13} + p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
19 1+6T−9T2−210T3−28pT4−900T5−2151T6+42756T7+442783T8+42756pT9−2151p2T10−900p3T11−28p5T12−210p5T13−9p6T14+6p7T15+p8T16 1 + 6 T - 9 T^{2} - 210 T^{3} - 28 p T^{4} - 900 T^{5} - 2151 T^{6} + 42756 T^{7} + 442783 T^{8} + 42756 p T^{9} - 2151 p^{2} T^{10} - 900 p^{3} T^{11} - 28 p^{5} T^{12} - 210 p^{5} T^{13} - 9 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
23 (1−4T−7T2+120T3−319T4+120pT5−7p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 7 T^{2} + 120 T^{3} - 319 T^{4} + 120 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 1−8T−2T2+312T3−1349T4−520T5+22492T6−95040T7+530997T8−95040pT9+22492p2T10−520p3T11−1349p4T12+312p5T13−2p6T14−8p7T15+p8T16 1 - 8 T - 2 T^{2} + 312 T^{3} - 1349 T^{4} - 520 T^{5} + 22492 T^{6} - 95040 T^{7} + 530997 T^{8} - 95040 p T^{9} + 22492 p^{2} T^{10} - 520 p^{3} T^{11} - 1349 p^{4} T^{12} + 312 p^{5} T^{13} - 2 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
37 (1−T+pT2)8 ( 1 - T + p T^{2} )^{8}
41 1+2T−7T2+46T3−1348T4+11740T5+53847T6−209308T7+2217735T8−209308pT9+53847p2T10+11740p3T11−1348p4T12+46p5T13−7p6T14+2p7T15+p8T16 1 + 2 T - 7 T^{2} + 46 T^{3} - 1348 T^{4} + 11740 T^{5} + 53847 T^{6} - 209308 T^{7} + 2217735 T^{8} - 209308 p T^{9} + 53847 p^{2} T^{10} + 11740 p^{3} T^{11} - 1348 p^{4} T^{12} + 46 p^{5} T^{13} - 7 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
43 1−2T+15T2−138T3−1236T4−6996T5−15471T6+315444T7+2085951T8+315444pT9−15471p2T10−6996p3T11−1236p4T12−138p5T13+15p6T14−2p7T15+p8T16 1 - 2 T + 15 T^{2} - 138 T^{3} - 1236 T^{4} - 6996 T^{5} - 15471 T^{6} + 315444 T^{7} + 2085951 T^{8} + 315444 p T^{9} - 15471 p^{2} T^{10} - 6996 p^{3} T^{11} - 1236 p^{4} T^{12} - 138 p^{5} T^{13} + 15 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
47 1+8T−14T2−360T3−989T4+35464T5+256420T6−291456T7−6813243T8−291456pT9+256420p2T10+35464p3T11−989p4T12−360p5T13−14p6T14+8p7T15+p8T16 1 + 8 T - 14 T^{2} - 360 T^{3} - 989 T^{4} + 35464 T^{5} + 256420 T^{6} - 291456 T^{7} - 6813243 T^{8} - 291456 p T^{9} + 256420 p^{2} T^{10} + 35464 p^{3} T^{11} - 989 p^{4} T^{12} - 360 p^{5} T^{13} - 14 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
53 1+6T−71T2−750T3+2196T4+15468T5−181225T6+62652T7+18517247T8+62652pT9−181225p2T10+15468p3T11+2196p4T12−750p5T13−71p6T14+6p7T15+p8T16 1 + 6 T - 71 T^{2} - 750 T^{3} + 2196 T^{4} + 15468 T^{5} - 181225 T^{6} + 62652 T^{7} + 18517247 T^{8} + 62652 p T^{9} - 181225 p^{2} T^{10} + 15468 p^{3} T^{11} + 2196 p^{4} T^{12} - 750 p^{5} T^{13} - 71 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
59 1−6T−41T2+6pT3−324T4−1092pT5+428921T6+1269852T7−16710673T8+1269852pT9+428921p2T10−1092p4T11−324p4T12+6p6T13−41p6T14−6p7T15+p8T16 1 - 6 T - 41 T^{2} + 6 p T^{3} - 324 T^{4} - 1092 p T^{5} + 428921 T^{6} + 1269852 T^{7} - 16710673 T^{8} + 1269852 p T^{9} + 428921 p^{2} T^{10} - 1092 p^{4} T^{11} - 324 p^{4} T^{12} + 6 p^{6} T^{13} - 41 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
61 (1+114T2+p2T4)4 ( 1 + 114 T^{2} + p^{2} T^{4} )^{4}
67 (1+2T+117T2+2pT3+p2T4)4 ( 1 + 2 T + 117 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4}
71 1−14T+55T2+210T3−1820T4−72772T5+789433T6−40740T7−29229705T8−40740pT9+789433p2T10−72772p3T11−1820p4T12+210p5T13+55p6T14−14p7T15+p8T16 1 - 14 T + 55 T^{2} + 210 T^{3} - 1820 T^{4} - 72772 T^{5} + 789433 T^{6} - 40740 T^{7} - 29229705 T^{8} - 40740 p T^{9} + 789433 p^{2} T^{10} - 72772 p^{3} T^{11} - 1820 p^{4} T^{12} + 210 p^{5} T^{13} + 55 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16}
73 1+2T−135T2−402T3+12924T4+15516T5−1104681T6−336924T7+86270151T8−336924pT9−1104681p2T10+15516p3T11+12924p4T12−402p5T13−135p6T14+2p7T15+p8T16 1 + 2 T - 135 T^{2} - 402 T^{3} + 12924 T^{4} + 15516 T^{5} - 1104681 T^{6} - 336924 T^{7} + 86270151 T^{8} - 336924 p T^{9} - 1104681 p^{2} T^{10} + 15516 p^{3} T^{11} + 12924 p^{4} T^{12} - 402 p^{5} T^{13} - 135 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
79 1−22T+223T2−902T3−6364T4+127820T5−975103T6+4781260T7−33119273T8+4781260pT9−975103p2T10+127820p3T11−6364p4T12−902p5T13+223p6T14−22p7T15+p8T16 1 - 22 T + 223 T^{2} - 902 T^{3} - 6364 T^{4} + 127820 T^{5} - 975103 T^{6} + 4781260 T^{7} - 33119273 T^{8} + 4781260 p T^{9} - 975103 p^{2} T^{10} + 127820 p^{3} T^{11} - 6364 p^{4} T^{12} - 902 p^{5} T^{13} + 223 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16}
83 1−6T−89T2+786T3+2508T4−114396T5+594569T6+4068348T7−59039617T8+4068348pT9+594569p2T10−114396p3T11+2508p4T12+786p5T13−89p6T14−6p7T15+p8T16 1 - 6 T - 89 T^{2} + 786 T^{3} + 2508 T^{4} - 114396 T^{5} + 594569 T^{6} + 4068348 T^{7} - 59039617 T^{8} + 4068348 p T^{9} + 594569 p^{2} T^{10} - 114396 p^{3} T^{11} + 2508 p^{4} T^{12} + 786 p^{5} T^{13} - 89 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
89 1−8T−58T2+728T3−973T4−183880T5+1530828T6+5113024T7−89035515T8+5113024pT9+1530828p2T10−183880p3T11−973p4T12+728p5T13−58p6T14−8p7T15+p8T16 1 - 8 T - 58 T^{2} + 728 T^{3} - 973 T^{4} - 183880 T^{5} + 1530828 T^{6} + 5113024 T^{7} - 89035515 T^{8} + 5113024 p T^{9} + 1530828 p^{2} T^{10} - 183880 p^{3} T^{11} - 973 p^{4} T^{12} + 728 p^{5} T^{13} - 58 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
97 1+16T+6T2−2352T3−23709T4−154992T5−259380T6+19931520T7+366279717T8+19931520pT9−259380p2T10−154992p3T11−23709p4T12−2352p5T13+6p6T14+16p7T15+p8T16 1 + 16 T + 6 T^{2} - 2352 T^{3} - 23709 T^{4} - 154992 T^{5} - 259380 T^{6} + 19931520 T^{7} + 366279717 T^{8} + 19931520 p T^{9} - 259380 p^{2} T^{10} - 154992 p^{3} T^{11} - 23709 p^{4} T^{12} - 2352 p^{5} T^{13} + 6 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} 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

−4.32720161370068551670048824374, −4.08955010155712923063884171444, −3.98710575417456850733211323723, −3.85061750855607533508741577162, −3.79059463834889436695612548690, −3.60382032288679335875853609184, −3.59575494352327543846446183484, −3.56075395288620057252995660896, −3.23480233077716382685637102929, −2.92807483716033048219941176739, −2.81616345419647021664091606383, −2.81392280268649604270826936679, −2.62000813359039061467415106170, −2.45818847637520460292236849325, −2.38464662748497733841810032137, −2.18525098302005431291556920029, −2.03349697075436065779046871356, −1.90863529545816944647210107330, −1.81709846590302760976797560754, −1.63030233871540320771753823454, −1.40290572056942639005611905454, −1.20261886653715004192517510789, −1.19749023660935613233197011027, −0.57987589386918786278358559211, −0.56796364114756866712860295698, 0.56796364114756866712860295698, 0.57987589386918786278358559211, 1.19749023660935613233197011027, 1.20261886653715004192517510789, 1.40290572056942639005611905454, 1.63030233871540320771753823454, 1.81709846590302760976797560754, 1.90863529545816944647210107330, 2.03349697075436065779046871356, 2.18525098302005431291556920029, 2.38464662748497733841810032137, 2.45818847637520460292236849325, 2.62000813359039061467415106170, 2.81392280268649604270826936679, 2.81616345419647021664091606383, 2.92807483716033048219941176739, 3.23480233077716382685637102929, 3.56075395288620057252995660896, 3.59575494352327543846446183484, 3.60382032288679335875853609184, 3.79059463834889436695612548690, 3.85061750855607533508741577162, 3.98710575417456850733211323723, 4.08955010155712923063884171444, 4.32720161370068551670048824374

Graph of the ZZ-function along the critical line

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