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Properties

Label 16-6000e8-1.1-c1e8-0-0
Degree 1616
Conductor 1.680×10301.680\times 10^{30}
Sign 11
Analytic cond. 2.77604×10132.77604\times 10^{13}
Root an. cond. 6.921726.92172
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  − 4·9-s − 6·11-s − 30·29-s − 12·31-s + 26·41-s + 9·49-s + 16·59-s + 26·61-s + 18·71-s + 42·79-s + 10·81-s − 24·89-s + 24·99-s + 62·101-s − 12·109-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s + 173-s + ⋯
L(s)  = 1  − 4/3·9-s − 1.80·11-s − 5.57·29-s − 2.15·31-s + 4.06·41-s + 9/7·49-s + 2.08·59-s + 3.32·61-s + 2.13·71-s + 4.72·79-s + 10/9·81-s − 2.54·89-s + 2.41·99-s + 6.16·101-s − 1.14·109-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((232⋅38⋅524)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅38⋅524)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 232⋅38⋅5242^{32} \cdot 3^{8} \cdot 5^{24}
Sign: 11
Analytic conductor: 2.77604×10132.77604\times 10^{13}
Root analytic conductor: 6.921726.92172
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 232⋅38⋅524, ( :[1/2]8), 1)(16,\ 2^{32} \cdot 3^{8} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.23402955970.2340295597
L(12)L(\frac12) ≈\approx 0.23402955970.2340295597
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
3 (1+T2)4 ( 1 + T^{2} )^{4}
5 1 1
good7 1−9T2+139T4−711T6+7704T8−711p2T10+139p4T12−9p6T14+p8T16 1 - 9 T^{2} + 139 T^{4} - 711 T^{6} + 7704 T^{8} - 711 p^{2} T^{10} + 139 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16}
11 (1+3T+20T2+54T3+243T4+54pT5+20p2T6+3p3T7+p4T8)2 ( 1 + 3 T + 20 T^{2} + 54 T^{3} + 243 T^{4} + 54 p T^{5} + 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
13 1−3pT2+790T4−10368T6+128673T8−10368p2T10+790p4T12−3p7T14+p8T16 1 - 3 p T^{2} + 790 T^{4} - 10368 T^{6} + 128673 T^{8} - 10368 p^{2} T^{10} + 790 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16}
17 (1−65T2+1633T4−65p2T6+p4T8)2 ( 1 - 65 T^{2} + 1633 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2}
19 (1+39T2−60T3+44pT4−60pT5+39p2T6+p4T8)2 ( 1 + 39 T^{2} - 60 T^{3} + 44 p T^{4} - 60 p T^{5} + 39 p^{2} T^{6} + p^{4} T^{8} )^{2}
23 1−123T2+7330T4−279288T6+7526469T8−279288p2T10+7330p4T12−123p6T14+p8T16 1 - 123 T^{2} + 7330 T^{4} - 279288 T^{6} + 7526469 T^{8} - 279288 p^{2} T^{10} + 7330 p^{4} T^{12} - 123 p^{6} T^{14} + p^{8} T^{16}
29 (1+15T+164T2+1170T3+7281T4+1170pT5+164p2T6+15p3T7+p4T8)2 ( 1 + 15 T + 164 T^{2} + 1170 T^{3} + 7281 T^{4} + 1170 p T^{5} + 164 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2}
31 (1+6T+3pT2+318T3+3539T4+318pT5+3p3T6+6p3T7+p4T8)2 ( 1 + 6 T + 3 p T^{2} + 318 T^{3} + 3539 T^{4} + 318 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
37 1−47T2+2670T4−90832T6+5094089T8−90832p2T10+2670p4T12−47p6T14+p8T16 1 - 47 T^{2} + 2670 T^{4} - 90832 T^{6} + 5094089 T^{8} - 90832 p^{2} T^{10} + 2670 p^{4} T^{12} - 47 p^{6} T^{14} + p^{8} T^{16}
41 (1−13T+155T2−1249T3+8848T4−1249pT5+155p2T6−13p3T7+p4T8)2 ( 1 - 13 T + 155 T^{2} - 1249 T^{3} + 8848 T^{4} - 1249 p T^{5} + 155 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 (1−125T2+7053T4−125p2T6+p4T8)2 ( 1 - 125 T^{2} + 7053 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} )^{2}
47 1−183T2+11962T4−248976T6−1203891T8−248976p2T10+11962p4T12−183p6T14+p8T16 1 - 183 T^{2} + 11962 T^{4} - 248976 T^{6} - 1203891 T^{8} - 248976 p^{2} T^{10} + 11962 p^{4} T^{12} - 183 p^{6} T^{14} + p^{8} T^{16}
53 1−301T2+40599T4−3376979T6+203717984T8−3376979p2T10+40599p4T12−301p6T14+p8T16 1 - 301 T^{2} + 40599 T^{4} - 3376979 T^{6} + 203717984 T^{8} - 3376979 p^{2} T^{10} + 40599 p^{4} T^{12} - 301 p^{6} T^{14} + p^{8} T^{16}
59 (1−8T+47T2+574T3−6221T4+574pT5+47p2T6−8p3T7+p4T8)2 ( 1 - 8 T + 47 T^{2} + 574 T^{3} - 6221 T^{4} + 574 p T^{5} + 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
61 (1−13T+121T2−703T3+3124T4−703pT5+121p2T6−13p3T7+p4T8)2 ( 1 - 13 T + 121 T^{2} - 703 T^{3} + 3124 T^{4} - 703 p T^{5} + 121 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2}
67 1+3T2+6415T4+99pT6+48836664T8+99p3T10+6415p4T12+3p6T14+p8T16 1 + 3 T^{2} + 6415 T^{4} + 99 p T^{6} + 48836664 T^{8} + 99 p^{3} T^{10} + 6415 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16}
71 (1−9T+283T2−1797T3+30024T4−1797pT5+283p2T6−9p3T7+p4T8)2 ( 1 - 9 T + 283 T^{2} - 1797 T^{3} + 30024 T^{4} - 1797 p T^{5} + 283 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 1−377T2+72387T4−8942539T6+772942784T8−8942539p2T10+72387p4T12−377p6T14+p8T16 1 - 377 T^{2} + 72387 T^{4} - 8942539 T^{6} + 772942784 T^{8} - 8942539 p^{2} T^{10} + 72387 p^{4} T^{12} - 377 p^{6} T^{14} + p^{8} T^{16}
79 (1−21T+319T2−3879T3+37884T4−3879pT5+319p2T6−21p3T7+p4T8)2 ( 1 - 21 T + 319 T^{2} - 3879 T^{3} + 37884 T^{4} - 3879 p T^{5} + 319 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2}
83 1−229T2+37515T4−49201pT6+385185608T8−49201p3T10+37515p4T12−229p6T14+p8T16 1 - 229 T^{2} + 37515 T^{4} - 49201 p T^{6} + 385185608 T^{8} - 49201 p^{3} T^{10} + 37515 p^{4} T^{12} - 229 p^{6} T^{14} + p^{8} T^{16}
89 (1+12T+353T2+3150T3+46956T4+3150pT5+353p2T6+12p3T7+p4T8)2 ( 1 + 12 T + 353 T^{2} + 3150 T^{3} + 46956 T^{4} + 3150 p T^{5} + 353 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 1−625T2+181591T4−32088775T6+3771831376T8−32088775p2T10+181591p4T12−625p6T14+p8T16 1 - 625 T^{2} + 181591 T^{4} - 32088775 T^{6} + 3771831376 T^{8} - 32088775 p^{2} T^{10} + 181591 p^{4} T^{12} - 625 p^{6} T^{14} + 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

−3.38861659025515183525242837786, −3.37653686068264132714555389966, −2.94893224289645595460302696307, −2.94496304969601059052646909845, −2.77643375271936565162704064517, −2.73116255750232733030998399361, −2.53255577114478525743253537281, −2.38742402314291200695983636424, −2.37461751427821487030254901585, −2.23202021655212948037062778437, −2.19193377324302399889635487629, −2.15152711860714224420826447826, −1.99403846333524756531823879467, −1.92954107625152772597088603176, −1.90832263229045716149693332363, −1.50577344870171157545923384560, −1.43692979273384505555012382582, −1.15439249352552819897464043570, −1.05984166767295396924635449964, −0.927146285051308016546901074780, −0.852858671187509984375719036122, −0.53931515961035545194060220009, −0.39729088829953576766156842326, −0.37256297523656360201148427452, −0.04250184961434522257182240598, 0.04250184961434522257182240598, 0.37256297523656360201148427452, 0.39729088829953576766156842326, 0.53931515961035545194060220009, 0.852858671187509984375719036122, 0.927146285051308016546901074780, 1.05984166767295396924635449964, 1.15439249352552819897464043570, 1.43692979273384505555012382582, 1.50577344870171157545923384560, 1.90832263229045716149693332363, 1.92954107625152772597088603176, 1.99403846333524756531823879467, 2.15152711860714224420826447826, 2.19193377324302399889635487629, 2.23202021655212948037062778437, 2.37461751427821487030254901585, 2.38742402314291200695983636424, 2.53255577114478525743253537281, 2.73116255750232733030998399361, 2.77643375271936565162704064517, 2.94496304969601059052646909845, 2.94893224289645595460302696307, 3.37653686068264132714555389966, 3.38861659025515183525242837786

Graph of the ZZ-function along the critical line

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