↑

Properties

Label 16-270e8-1.1-c2e8-0-3
Degree 1616
Conductor 2.824×10192.824\times 10^{19}
Sign 11
Analytic cond. 8.58203×1068.58203\times 10^{6}
Root an. cond. 2.712372.71237
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 32·4-s + 12·5-s − 8·7-s − 80·8-s − 96·10-s − 40·11-s − 24·13-s + 64·14-s + 120·16-s + 8·17-s + 384·20-s + 320·22-s − 16·23-s + 80·25-s + 192·26-s − 256·28-s − 64·31-s − 32·32-s − 64·34-s − 96·35-s − 24·37-s − 960·40-s + 56·41-s − 72·43-s − 1.28e3·44-s + 128·46-s + ⋯
L(s)  = 1  − 4·2-s + 8·4-s + 12/5·5-s − 8/7·7-s − 10·8-s − 9.59·10-s − 3.63·11-s − 1.84·13-s + 32/7·14-s + 15/2·16-s + 8/17·17-s + 96/5·20-s + 14.5·22-s − 0.695·23-s + 16/5·25-s + 7.38·26-s − 9.14·28-s − 2.06·31-s − 32-s − 1.88·34-s − 2.74·35-s − 0.648·37-s − 24·40-s + 1.36·41-s − 1.67·43-s − 29.0·44-s + 2.78·46-s + ⋯

Functional equation

Λ(s)=((28⋅324⋅58)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((28⋅324⋅58)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅324⋅582^{8} \cdot 3^{24} \cdot 5^{8}
Sign: 11
Analytic conductor: 8.58203×1068.58203\times 10^{6}
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅324⋅58, ( :[1]8), 1)(16,\ 2^{8} \cdot 3^{24} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.27484551100.2748455110
L(12)L(\frac12) ≈\approx 0.27484551100.2748455110
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+pT+pT2)4 ( 1 + p T + p T^{2} )^{4}
3 1 1
5 1−12T+64T2−108T3−24pT4−108p2T5+64p4T6−12p6T7+p8T8 1 - 12 T + 64 T^{2} - 108 T^{3} - 24 p T^{4} - 108 p^{2} T^{5} + 64 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8}
good7 1+8T+32T2+256T3−3442T4−27464T5−76800T6−207864T7+5077843T8−207864p2T9−76800p4T10−27464p6T11−3442p8T12+256p10T13+32p12T14+8p14T15+p16T16 1 + 8 T + 32 T^{2} + 256 T^{3} - 3442 T^{4} - 27464 T^{5} - 76800 T^{6} - 207864 T^{7} + 5077843 T^{8} - 207864 p^{2} T^{9} - 76800 p^{4} T^{10} - 27464 p^{6} T^{11} - 3442 p^{8} T^{12} + 256 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16}
11 (1+20T+400T2+4340T3+55768T4+4340p2T5+400p4T6+20p6T7+p8T8)2 ( 1 + 20 T + 400 T^{2} + 4340 T^{3} + 55768 T^{4} + 4340 p^{2} T^{5} + 400 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2}
13 1+24T+288T2+5856T3+3622pT4−417240T5−6428160T6−165397320T7−3701745165T8−165397320p2T9−6428160p4T10−417240p6T11+3622p9T12+5856p10T13+288p12T14+24p14T15+p16T16 1 + 24 T + 288 T^{2} + 5856 T^{3} + 3622 p T^{4} - 417240 T^{5} - 6428160 T^{6} - 165397320 T^{7} - 3701745165 T^{8} - 165397320 p^{2} T^{9} - 6428160 p^{4} T^{10} - 417240 p^{6} T^{11} + 3622 p^{9} T^{12} + 5856 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16}
17 1−8T+32T2+7832T3−71596T4−403384T5+36188256T6+64357608T7+158386534T8+64357608p2T9+36188256p4T10−403384p6T11−71596p8T12+7832p10T13+32p12T14−8p14T15+p16T16 1 - 8 T + 32 T^{2} + 7832 T^{3} - 71596 T^{4} - 403384 T^{5} + 36188256 T^{6} + 64357608 T^{7} + 158386534 T^{8} + 64357608 p^{2} T^{9} + 36188256 p^{4} T^{10} - 403384 p^{6} T^{11} - 71596 p^{8} T^{12} + 7832 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16}
19 1−580T2+475930T4−192894064T6+91634293315T8−192894064p4T10+475930p8T12−580p12T14+p16T16 1 - 580 T^{2} + 475930 T^{4} - 192894064 T^{6} + 91634293315 T^{8} - 192894064 p^{4} T^{10} + 475930 p^{8} T^{12} - 580 p^{12} T^{14} + p^{16} T^{16}
23 1+16T+128T2+10040T3+228752T4−2307400T5−15797856T6−601862832T7−3438620990T8−601862832p2T9−15797856p4T10−2307400p6T11+228752p8T12+10040p10T13+128p12T14+16p14T15+p16T16 1 + 16 T + 128 T^{2} + 10040 T^{3} + 228752 T^{4} - 2307400 T^{5} - 15797856 T^{6} - 601862832 T^{7} - 3438620990 T^{8} - 601862832 p^{2} T^{9} - 15797856 p^{4} T^{10} - 2307400 p^{6} T^{11} + 228752 p^{8} T^{12} + 10040 p^{10} T^{13} + 128 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16}
29 1−3760T2+7513360T4−10124362384T6+9927642064450T8−10124362384p4T10+7513360p8T12−3760p12T14+p16T16 1 - 3760 T^{2} + 7513360 T^{4} - 10124362384 T^{6} + 9927642064450 T^{8} - 10124362384 p^{4} T^{10} + 7513360 p^{8} T^{12} - 3760 p^{12} T^{14} + p^{16} T^{16}
31 (1+32T+1384T2−19744T3−216494T4−19744p2T5+1384p4T6+32p6T7+p8T8)2 ( 1 + 32 T + 1384 T^{2} - 19744 T^{3} - 216494 T^{4} - 19744 p^{2} T^{5} + 1384 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2}
37 1+24T+288T2+62016T3−1193810T4−54832824T5+950821632T6−7540071336T7+489879159603T8−7540071336p2T9+950821632p4T10−54832824p6T11−1193810p8T12+62016p10T13+288p12T14+24p14T15+p16T16 1 + 24 T + 288 T^{2} + 62016 T^{3} - 1193810 T^{4} - 54832824 T^{5} + 950821632 T^{6} - 7540071336 T^{7} + 489879159603 T^{8} - 7540071336 p^{2} T^{9} + 950821632 p^{4} T^{10} - 54832824 p^{6} T^{11} - 1193810 p^{8} T^{12} + 62016 p^{10} T^{13} + 288 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16}
41 (1−28T+4480T2−2372pT3+9455128T4−2372p3T5+4480p4T6−28p6T7+p8T8)2 ( 1 - 28 T + 4480 T^{2} - 2372 p T^{3} + 9455128 T^{4} - 2372 p^{3} T^{5} + 4480 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2}
43 1+72T+2592T2+39960T3+5556412T4+548106696T5+25859863008T6+705930936408T7+15548175298374T8+705930936408p2T9+25859863008p4T10+548106696p6T11+5556412p8T12+39960p10T13+2592p12T14+72p14T15+p16T16 1 + 72 T + 2592 T^{2} + 39960 T^{3} + 5556412 T^{4} + 548106696 T^{5} + 25859863008 T^{6} + 705930936408 T^{7} + 15548175298374 T^{8} + 705930936408 p^{2} T^{9} + 25859863008 p^{4} T^{10} + 548106696 p^{6} T^{11} + 5556412 p^{8} T^{12} + 39960 p^{10} T^{13} + 2592 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16}
47 1+124416T3+6072004T4+373248pT5+7739670528T6+680045663232T7+6691298905734T8+680045663232p2T9+7739670528p4T10+373248p7T11+6072004p8T12+124416p10T13+p16T16 1 + 124416 T^{3} + 6072004 T^{4} + 373248 p T^{5} + 7739670528 T^{6} + 680045663232 T^{7} + 6691298905734 T^{8} + 680045663232 p^{2} T^{9} + 7739670528 p^{4} T^{10} + 373248 p^{7} T^{11} + 6072004 p^{8} T^{12} + 124416 p^{10} T^{13} + p^{16} T^{16}
53 1+80T+3200T2+292120T3+10074896T4−402405800T5−21765084000T6−2335299106800T7−229685261539070T8−2335299106800p2T9−21765084000p4T10−402405800p6T11+10074896p8T12+292120p10T13+3200p12T14+80p14T15+p16T16 1 + 80 T + 3200 T^{2} + 292120 T^{3} + 10074896 T^{4} - 402405800 T^{5} - 21765084000 T^{6} - 2335299106800 T^{7} - 229685261539070 T^{8} - 2335299106800 p^{2} T^{9} - 21765084000 p^{4} T^{10} - 402405800 p^{6} T^{11} + 10074896 p^{8} T^{12} + 292120 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16}
59 1−14008T2+100975756T4−492995752648T6+1888076779441894T8−492995752648p4T10+100975756p8T12−14008p12T14+p16T16 1 - 14008 T^{2} + 100975756 T^{4} - 492995752648 T^{6} + 1888076779441894 T^{8} - 492995752648 p^{4} T^{10} + 100975756 p^{8} T^{12} - 14008 p^{12} T^{14} + p^{16} T^{16}
61 (1−108T+12490T2−969552T3+68045523T4−969552p2T5+12490p4T6−108p6T7+p8T8)2 ( 1 - 108 T + 12490 T^{2} - 969552 T^{3} + 68045523 T^{4} - 969552 p^{2} T^{5} + 12490 p^{4} T^{6} - 108 p^{6} T^{7} + p^{8} T^{8} )^{2}
67 1−136T+9248T2−705536T3+33707342T4+897952072T5−184956456960T6+22832526785592T7−2128448215323821T8+22832526785592p2T9−184956456960p4T10+897952072p6T11+33707342p8T12−705536p10T13+9248p12T14−136p14T15+p16T16 1 - 136 T + 9248 T^{2} - 705536 T^{3} + 33707342 T^{4} + 897952072 T^{5} - 184956456960 T^{6} + 22832526785592 T^{7} - 2128448215323821 T^{8} + 22832526785592 p^{2} T^{9} - 184956456960 p^{4} T^{10} + 897952072 p^{6} T^{11} + 33707342 p^{8} T^{12} - 705536 p^{10} T^{13} + 9248 p^{12} T^{14} - 136 p^{14} T^{15} + p^{16} T^{16}
71 (1+28T+16048T2+231316T3+109038184T4+231316p2T5+16048p4T6+28p6T7+p8T8)2 ( 1 + 28 T + 16048 T^{2} + 231316 T^{3} + 109038184 T^{4} + 231316 p^{2} T^{5} + 16048 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2}
73 1−368T+67712T2−8839312T3+943013726T4−87425766400T5+7386054936960T6−587580358449120T7+44199309763992355T8−587580358449120p2T9+7386054936960p4T10−87425766400p6T11+943013726p8T12−8839312p10T13+67712p12T14−368p14T15+p16T16 1 - 368 T + 67712 T^{2} - 8839312 T^{3} + 943013726 T^{4} - 87425766400 T^{5} + 7386054936960 T^{6} - 587580358449120 T^{7} + 44199309763992355 T^{8} - 587580358449120 p^{2} T^{9} + 7386054936960 p^{4} T^{10} - 87425766400 p^{6} T^{11} + 943013726 p^{8} T^{12} - 8839312 p^{10} T^{13} + 67712 p^{12} T^{14} - 368 p^{14} T^{15} + p^{16} T^{16}
79 1−25340T2+370619530T4−3658708310384T6+26467945053296275T8−3658708310384p4T10+370619530p8T12−25340p12T14+p16T16 1 - 25340 T^{2} + 370619530 T^{4} - 3658708310384 T^{6} + 26467945053296275 T^{8} - 3658708310384 p^{4} T^{10} + 370619530 p^{8} T^{12} - 25340 p^{12} T^{14} + p^{16} T^{16}
83 1−168T+14112T2−916800T3−16348688T4+4879673856T5−168811402752T6−16030525612392T7+3571925738115714T8−16030525612392p2T9−168811402752p4T10+4879673856p6T11−16348688p8T12−916800p10T13+14112p12T14−168p14T15+p16T16 1 - 168 T + 14112 T^{2} - 916800 T^{3} - 16348688 T^{4} + 4879673856 T^{5} - 168811402752 T^{6} - 16030525612392 T^{7} + 3571925738115714 T^{8} - 16030525612392 p^{2} T^{9} - 168811402752 p^{4} T^{10} + 4879673856 p^{6} T^{11} - 16348688 p^{8} T^{12} - 916800 p^{10} T^{13} + 14112 p^{12} T^{14} - 168 p^{14} T^{15} + p^{16} T^{16}
89 1−34744T2+427201228T4−1863800468296T6+2580261126216742T8−1863800468296p4T10+427201228p8T12−34744p12T14+p16T16 1 - 34744 T^{2} + 427201228 T^{4} - 1863800468296 T^{6} + 2580261126216742 T^{8} - 1863800468296 p^{4} T^{10} + 427201228 p^{8} T^{12} - 34744 p^{12} T^{14} + p^{16} T^{16}
97 1+312T+48672T2+6967680T3+1062851086T4+133636973544T5+14237929979136T6+1553115219249528T7+162662200396802835T8+1553115219249528p2T9+14237929979136p4T10+133636973544p6T11+1062851086p8T12+6967680p10T13+48672p12T14+312p14T15+p16T16 1 + 312 T + 48672 T^{2} + 6967680 T^{3} + 1062851086 T^{4} + 133636973544 T^{5} + 14237929979136 T^{6} + 1553115219249528 T^{7} + 162662200396802835 T^{8} + 1553115219249528 p^{2} T^{9} + 14237929979136 p^{4} T^{10} + 133636973544 p^{6} T^{11} + 1062851086 p^{8} T^{12} + 6967680 p^{10} T^{13} + 48672 p^{12} T^{14} + 312 p^{14} T^{15} + p^{16} T^{16}
show more
show less
   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

−5.22484041655693816929755836648, −5.21163872928654980305866265414, −4.76584703474072910500199190211, −4.75543939025247428031016259734, −4.69657934892309397250991694702, −4.50792331068814571894094624087, −4.20038646135608457859117084791, −3.78046633505642825787948939473, −3.69857492196017408664227125299, −3.53334088993581534931794482628, −3.33220308286275044153637287016, −3.18139463178982154775879454087, −2.74639128598358044864235028200, −2.66993606969624752926930020673, −2.61131573476276352897702861920, −2.34437271974939728226819702891, −2.15557169184256291185029624561, −2.02281418403474007643836066484, −2.01000971734080356047765334173, −1.59608802808158665060268182157, −1.52903910314353430013784854878, −0.826156839722551317834885956186, −0.67161685326983419943729563936, −0.40016101647461352776988988529, −0.30464754717572462909757076254, 0.30464754717572462909757076254, 0.40016101647461352776988988529, 0.67161685326983419943729563936, 0.826156839722551317834885956186, 1.52903910314353430013784854878, 1.59608802808158665060268182157, 2.01000971734080356047765334173, 2.02281418403474007643836066484, 2.15557169184256291185029624561, 2.34437271974939728226819702891, 2.61131573476276352897702861920, 2.66993606969624752926930020673, 2.74639128598358044864235028200, 3.18139463178982154775879454087, 3.33220308286275044153637287016, 3.53334088993581534931794482628, 3.69857492196017408664227125299, 3.78046633505642825787948939473, 4.20038646135608457859117084791, 4.50792331068814571894094624087, 4.69657934892309397250991694702, 4.75543939025247428031016259734, 4.76584703474072910500199190211, 5.21163872928654980305866265414, 5.22484041655693816929755836648

Graph of the ZZ-function along the critical line

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