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Properties

Label 16-416e8-1.1-c1e8-0-4
Degree 1616
Conductor 8.969×10208.969\times 10^{20}
Sign 11
Analytic cond. 14823.914823.9
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 4·5-s − 2·7-s + 14·9-s − 6·11-s + 24·13-s − 24·15-s + 12·17-s + 6·19-s + 12·21-s − 6·23-s + 8·25-s − 12·27-s − 12·31-s + 36·33-s − 8·35-s − 144·39-s − 4·41-s − 22·43-s + 56·45-s + 8·47-s + 2·49-s − 72·51-s − 48·53-s − 24·55-s − 36·57-s + 22·59-s + ⋯
L(s)  = 1  − 3.46·3-s + 1.78·5-s − 0.755·7-s + 14/3·9-s − 1.80·11-s + 6.65·13-s − 6.19·15-s + 2.91·17-s + 1.37·19-s + 2.61·21-s − 1.25·23-s + 8/5·25-s − 2.30·27-s − 2.15·31-s + 6.26·33-s − 1.35·35-s − 23.0·39-s − 0.624·41-s − 3.35·43-s + 8.34·45-s + 1.16·47-s + 2/7·49-s − 10.0·51-s − 6.59·53-s − 3.23·55-s − 4.76·57-s + 2.86·59-s + ⋯

Functional equation

Λ(s)=((240⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((240⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 240⋅1382^{40} \cdot 13^{8}
Sign: 11
Analytic conductor: 14823.914823.9
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 240⋅138, ( :[1/2]8), 1)(16,\ 2^{40} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.2800317371.280031737
L(12)L(\frac12) ≈\approx 1.2800317371.280031737
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)
bad2 1 1
13 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
good3 1+2pT+22T2+20pT3+43pT4+28p2T5+466T6+94p2T7+1516T8+94p3T9+466p2T10+28p5T11+43p5T12+20p6T13+22p6T14+2p8T15+p8T16 1 + 2 p T + 22 T^{2} + 20 p T^{3} + 43 p T^{4} + 28 p^{2} T^{5} + 466 T^{6} + 94 p^{2} T^{7} + 1516 T^{8} + 94 p^{3} T^{9} + 466 p^{2} T^{10} + 28 p^{5} T^{11} + 43 p^{5} T^{12} + 20 p^{6} T^{13} + 22 p^{6} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16}
5 1−4T+8T2−12T3−4T4+76T5−8p2T6+84pT7−794T8+84p2T9−8p4T10+76p3T11−4p4T12−12p5T13+8p6T14−4p7T15+p8T16 1 - 4 T + 8 T^{2} - 12 T^{3} - 4 T^{4} + 76 T^{5} - 8 p^{2} T^{6} + 84 p T^{7} - 794 T^{8} + 84 p^{2} T^{9} - 8 p^{4} T^{10} + 76 p^{3} T^{11} - 4 p^{4} T^{12} - 12 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
7 1+2T+2T2+4pT3+101T4+4p2T5+582T6+2202T7+5716T8+2202pT9+582p2T10+4p5T11+101p4T12+4p6T13+2p6T14+2p7T15+p8T16 1 + 2 T + 2 T^{2} + 4 p T^{3} + 101 T^{4} + 4 p^{2} T^{5} + 582 T^{6} + 2202 T^{7} + 5716 T^{8} + 2202 p T^{9} + 582 p^{2} T^{10} + 4 p^{5} T^{11} + 101 p^{4} T^{12} + 4 p^{6} T^{13} + 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
11 1+6T+30T2+84T3+145T4−348T5−3030T6−17202T7−56916T8−17202pT9−3030p2T10−348p3T11+145p4T12+84p5T13+30p6T14+6p7T15+p8T16 1 + 6 T + 30 T^{2} + 84 T^{3} + 145 T^{4} - 348 T^{5} - 3030 T^{6} - 17202 T^{7} - 56916 T^{8} - 17202 p T^{9} - 3030 p^{2} T^{10} - 348 p^{3} T^{11} + 145 p^{4} T^{12} + 84 p^{5} T^{13} + 30 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
17 1−12T+114T2−792T3+4693T4−23520T5+110490T6−471996T7+1997436T8−471996pT9+110490p2T10−23520p3T11+4693p4T12−792p5T13+114p6T14−12p7T15+p8T16 1 - 12 T + 114 T^{2} - 792 T^{3} + 4693 T^{4} - 23520 T^{5} + 110490 T^{6} - 471996 T^{7} + 1997436 T^{8} - 471996 p T^{9} + 110490 p^{2} T^{10} - 23520 p^{3} T^{11} + 4693 p^{4} T^{12} - 792 p^{5} T^{13} + 114 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16}
19 1−6T+30T2−252T3+1489T4−7620T5+40746T6−188694T7+823116T8−188694pT9+40746p2T10−7620p3T11+1489p4T12−252p5T13+30p6T14−6p7T15+p8T16 1 - 6 T + 30 T^{2} - 252 T^{3} + 1489 T^{4} - 7620 T^{5} + 40746 T^{6} - 188694 T^{7} + 823116 T^{8} - 188694 p T^{9} + 40746 p^{2} T^{10} - 7620 p^{3} T^{11} + 1489 p^{4} T^{12} - 252 p^{5} T^{13} + 30 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
23 1+6T−2T2−156T3−1151T4−5604T5−4262T6+135054T7+1066684T8+135054pT9−4262p2T10−5604p3T11−1151p4T12−156p5T13−2p6T14+6p7T15+p8T16 1 + 6 T - 2 T^{2} - 156 T^{3} - 1151 T^{4} - 5604 T^{5} - 4262 T^{6} + 135054 T^{7} + 1066684 T^{8} + 135054 p T^{9} - 4262 p^{2} T^{10} - 5604 p^{3} T^{11} - 1151 p^{4} T^{12} - 156 p^{5} T^{13} - 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
29 1−38T2+144T3−59T4−4824T5+11986T6+50472T7+186988T8+50472pT9+11986p2T10−4824p3T11−59p4T12+144p5T13−38p6T14+p8T16 1 - 38 T^{2} + 144 T^{3} - 59 T^{4} - 4824 T^{5} + 11986 T^{6} + 50472 T^{7} + 186988 T^{8} + 50472 p T^{9} + 11986 p^{2} T^{10} - 4824 p^{3} T^{11} - 59 p^{4} T^{12} + 144 p^{5} T^{13} - 38 p^{6} T^{14} + p^{8} T^{16}
31 1+12T+72T2+540T3+5500T4+36684T5+190008T6+1281084T7+8521734T8+1281084pT9+190008p2T10+36684p3T11+5500p4T12+540p5T13+72p6T14+12p7T15+p8T16 1 + 12 T + 72 T^{2} + 540 T^{3} + 5500 T^{4} + 36684 T^{5} + 190008 T^{6} + 1281084 T^{7} + 8521734 T^{8} + 1281084 p T^{9} + 190008 p^{2} T^{10} + 36684 p^{3} T^{11} + 5500 p^{4} T^{12} + 540 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
37 1+90T2−96T3+2929T4−12480T5+25218T6−933600T7−881004T8−933600pT9+25218p2T10−12480p3T11+2929p4T12−96p5T13+90p6T14+p8T16 1 + 90 T^{2} - 96 T^{3} + 2929 T^{4} - 12480 T^{5} + 25218 T^{6} - 933600 T^{7} - 881004 T^{8} - 933600 p T^{9} + 25218 p^{2} T^{10} - 12480 p^{3} T^{11} + 2929 p^{4} T^{12} - 96 p^{5} T^{13} + 90 p^{6} T^{14} + p^{8} T^{16}
41 1+4T−10T2−240T3−1339T4−2584T5+24694T6+292068T7−192452T8+292068pT9+24694p2T10−2584p3T11−1339p4T12−240p5T13−10p6T14+4p7T15+p8T16 1 + 4 T - 10 T^{2} - 240 T^{3} - 1339 T^{4} - 2584 T^{5} + 24694 T^{6} + 292068 T^{7} - 192452 T^{8} + 292068 p T^{9} + 24694 p^{2} T^{10} - 2584 p^{3} T^{11} - 1339 p^{4} T^{12} - 240 p^{5} T^{13} - 10 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
43 1+22T+150T2+788T3+14033T4+120588T5+419410T6+4311526T7+47947500T8+4311526pT9+419410p2T10+120588p3T11+14033p4T12+788p5T13+150p6T14+22p7T15+p8T16 1 + 22 T + 150 T^{2} + 788 T^{3} + 14033 T^{4} + 120588 T^{5} + 419410 T^{6} + 4311526 T^{7} + 47947500 T^{8} + 4311526 p T^{9} + 419410 p^{2} T^{10} + 120588 p^{3} T^{11} + 14033 p^{4} T^{12} + 788 p^{5} T^{13} + 150 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16}
47 (1−4T+8T2−100T3+766T4−100pT5+8p2T6−4p3T7+p4T8)2 ( 1 - 4 T + 8 T^{2} - 100 T^{3} + 766 T^{4} - 100 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 (1+24T+368T2+72pT3+31806T4+72p2T5+368p2T6+24p3T7+p4T8)2 ( 1 + 24 T + 368 T^{2} + 72 p T^{3} + 31806 T^{4} + 72 p^{2} T^{5} + 368 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2}
59 1−22T+410T2−4596T3+46661T4−322124T5+2140414T6−9312390T7+69548260T8−9312390pT9+2140414p2T10−322124p3T11+46661p4T12−4596p5T13+410p6T14−22p7T15+p8T16 1 - 22 T + 410 T^{2} - 4596 T^{3} + 46661 T^{4} - 322124 T^{5} + 2140414 T^{6} - 9312390 T^{7} + 69548260 T^{8} - 9312390 p T^{9} + 2140414 p^{2} T^{10} - 322124 p^{3} T^{11} + 46661 p^{4} T^{12} - 4596 p^{5} T^{13} + 410 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16}
61 1−4T−138T2+472T3+8177T4−10056T5−748634T6−248860T7+65687988T8−248860pT9−748634p2T10−10056p3T11+8177p4T12+472p5T13−138p6T14−4p7T15+p8T16 1 - 4 T - 138 T^{2} + 472 T^{3} + 8177 T^{4} - 10056 T^{5} - 748634 T^{6} - 248860 T^{7} + 65687988 T^{8} - 248860 p T^{9} - 748634 p^{2} T^{10} - 10056 p^{3} T^{11} + 8177 p^{4} T^{12} + 472 p^{5} T^{13} - 138 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
67 1−2T−22T2+596T3−955T4−1540T5+155118T6−1149330T7+1782532T8−1149330pT9+155118p2T10−1540p3T11−955p4T12+596p5T13−22p6T14−2p7T15+p8T16 1 - 2 T - 22 T^{2} + 596 T^{3} - 955 T^{4} - 1540 T^{5} + 155118 T^{6} - 1149330 T^{7} + 1782532 T^{8} - 1149330 p T^{9} + 155118 p^{2} T^{10} - 1540 p^{3} T^{11} - 955 p^{4} T^{12} + 596 p^{5} T^{13} - 22 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
71 1+14T+242T2+1956T3+15989T4−3428T5−834218T6−18859242T7−160743884T8−18859242pT9−834218p2T10−3428p3T11+15989p4T12+1956p5T13+242p6T14+14p7T15+p8T16 1 + 14 T + 242 T^{2} + 1956 T^{3} + 15989 T^{4} - 3428 T^{5} - 834218 T^{6} - 18859242 T^{7} - 160743884 T^{8} - 18859242 p T^{9} - 834218 p^{2} T^{10} - 3428 p^{3} T^{11} + 15989 p^{4} T^{12} + 1956 p^{5} T^{13} + 242 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
73 1+12T+72T2−396T3−7076T4−25668T5+279864T6+1767300T7+23147910T8+1767300pT9+279864p2T10−25668p3T11−7076p4T12−396p5T13+72p6T14+12p7T15+p8T16 1 + 12 T + 72 T^{2} - 396 T^{3} - 7076 T^{4} - 25668 T^{5} + 279864 T^{6} + 1767300 T^{7} + 23147910 T^{8} + 1767300 p T^{9} + 279864 p^{2} T^{10} - 25668 p^{3} T^{11} - 7076 p^{4} T^{12} - 396 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
79 1−224T2+30940T4−3015968T6+249183430T8−3015968p2T10+30940p4T12−224p6T14+p8T16 1 - 224 T^{2} + 30940 T^{4} - 3015968 T^{6} + 249183430 T^{8} - 3015968 p^{2} T^{10} + 30940 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16}
83 1−44T+968T2−15660T3+206492T4−2233516T5+21008248T6−186069324T7+1657961446T8−186069324pT9+21008248p2T10−2233516p3T11+206492p4T12−15660p5T13+968p6T14−44p7T15+p8T16 1 - 44 T + 968 T^{2} - 15660 T^{3} + 206492 T^{4} - 2233516 T^{5} + 21008248 T^{6} - 186069324 T^{7} + 1657961446 T^{8} - 186069324 p T^{9} + 21008248 p^{2} T^{10} - 2233516 p^{3} T^{11} + 206492 p^{4} T^{12} - 15660 p^{5} T^{13} + 968 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16}
89 1+32T+458T2+1848T3−39487T4−715376T5−3565454T6+33394392T7+654295780T8+33394392pT9−3565454p2T10−715376p3T11−39487p4T12+1848p5T13+458p6T14+32p7T15+p8T16 1 + 32 T + 458 T^{2} + 1848 T^{3} - 39487 T^{4} - 715376 T^{5} - 3565454 T^{6} + 33394392 T^{7} + 654295780 T^{8} + 33394392 p T^{9} - 3565454 p^{2} T^{10} - 715376 p^{3} T^{11} - 39487 p^{4} T^{12} + 1848 p^{5} T^{13} + 458 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16}
97 1−68T+1862T2−22264T3−16651T4+3698264T5−33762042T6−152904156T7+4581782908T8−152904156pT9−33762042p2T10+3698264p3T11−16651p4T12−22264p5T13+1862p6T14−68p7T15+p8T16 1 - 68 T + 1862 T^{2} - 22264 T^{3} - 16651 T^{4} + 3698264 T^{5} - 33762042 T^{6} - 152904156 T^{7} + 4581782908 T^{8} - 152904156 p T^{9} - 33762042 p^{2} T^{10} + 3698264 p^{3} T^{11} - 16651 p^{4} T^{12} - 22264 p^{5} T^{13} + 1862 p^{6} T^{14} - 68 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

−5.19070343920528765683141300868, −4.95985794968096808861676130563, −4.84626610922084159777285236790, −4.78108041171022257978986034173, −4.59100384857136812410330363218, −4.25455343496039020146304197730, −4.02205998338317954254524574932, −3.94089780944467549905547808420, −3.58005204612683143344344336354, −3.57258443076229461536976002561, −3.38910595712710684130421419801, −3.33025575084907479273667016818, −3.28973872724973852447483416302, −3.14632230242505281418508542330, −3.12220261457826960377958997611, −2.65912322804041651295669807508, −2.13800153624399432219774384029, −1.97036190305330429753729176399, −1.95329255781088577836564254144, −1.71773422448241610247812015253, −1.32966464616894399132730820979, −1.26890940056556601935506965929, −1.07862855370893804093966555214, −0.66876770173187298434773279440, −0.39861124710530090784570850206, 0.39861124710530090784570850206, 0.66876770173187298434773279440, 1.07862855370893804093966555214, 1.26890940056556601935506965929, 1.32966464616894399132730820979, 1.71773422448241610247812015253, 1.95329255781088577836564254144, 1.97036190305330429753729176399, 2.13800153624399432219774384029, 2.65912322804041651295669807508, 3.12220261457826960377958997611, 3.14632230242505281418508542330, 3.28973872724973852447483416302, 3.33025575084907479273667016818, 3.38910595712710684130421419801, 3.57258443076229461536976002561, 3.58005204612683143344344336354, 3.94089780944467549905547808420, 4.02205998338317954254524574932, 4.25455343496039020146304197730, 4.59100384857136812410330363218, 4.78108041171022257978986034173, 4.84626610922084159777285236790, 4.95985794968096808861676130563, 5.19070343920528765683141300868

Graph of the ZZ-function along the critical line

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