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Properties

Label 16-2640e8-1.1-c1e8-0-4
Degree 1616
Conductor 2.360×10272.360\times 10^{27}
Sign 11
Analytic cond. 3.89985×10103.89985\times 10^{10}
Root an. cond. 4.591354.59135
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  − 2·3-s + 3·9-s − 6·11-s + 4·17-s − 4·25-s + 4·27-s − 4·29-s + 4·31-s + 12·33-s − 12·37-s − 16·41-s + 26·49-s − 8·51-s + 12·67-s + 8·75-s − 11·81-s + 72·83-s + 8·87-s − 8·93-s − 8·97-s − 18·99-s + 52·101-s + 8·103-s + 24·111-s + 24·121-s + 32·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.80·11-s + 0.970·17-s − 4/5·25-s + 0.769·27-s − 0.742·29-s + 0.718·31-s + 2.08·33-s − 1.97·37-s − 2.49·41-s + 26/7·49-s − 1.12·51-s + 1.46·67-s + 0.923·75-s − 1.22·81-s + 7.90·83-s + 0.857·87-s − 0.829·93-s − 0.812·97-s − 1.80·99-s + 5.17·101-s + 0.788·103-s + 2.27·111-s + 2.18·121-s + 2.88·123-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=((232⋅38⋅58⋅118)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅38⋅58⋅118)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^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 232⋅38⋅58⋅1182^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}
Sign: 11
Analytic conductor: 3.89985×10103.89985\times 10^{10}
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 232⋅38⋅58⋅118, ( :[1/2]8), 1)(16,\ 2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.9780089931.978008993
L(12)L(\frac12) ≈\approx 1.9780089931.978008993
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+2T+T2−8T3−16T4−8pT5+p2T6+2p3T7+p4T8 1 + 2 T + T^{2} - 8 T^{3} - 16 T^{4} - 8 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
5 (1+T2)4 ( 1 + T^{2} )^{4}
11 1+6T+12T2−6pT3−378T4−6p2T5+12p2T6+6p3T7+p4T8 1 + 6 T + 12 T^{2} - 6 p T^{3} - 378 T^{4} - 6 p^{2} T^{5} + 12 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
good7 1−26T2+47pT4−3098T6+24180T8−3098p2T10+47p5T12−26p6T14+p8T16 1 - 26 T^{2} + 47 p T^{4} - 3098 T^{6} + 24180 T^{8} - 3098 p^{2} T^{10} + 47 p^{5} T^{12} - 26 p^{6} T^{14} + p^{8} T^{16}
13 1−60T2+1892T4−39972T6+607318T8−39972p2T10+1892p4T12−60p6T14+p8T16 1 - 60 T^{2} + 1892 T^{4} - 39972 T^{6} + 607318 T^{8} - 39972 p^{2} T^{10} + 1892 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16}
17 (1−2T+37T2+2T3+36pT4+2pT5+37p2T6−2p3T7+p4T8)2 ( 1 - 2 T + 37 T^{2} + 2 T^{3} + 36 p T^{4} + 2 p T^{5} + 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
19 1−54T2+1513T4−24406T6+389156T8−24406p2T10+1513p4T12−54p6T14+p8T16 1 - 54 T^{2} + 1513 T^{4} - 24406 T^{6} + 389156 T^{8} - 24406 p^{2} T^{10} + 1513 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16}
23 1−80T2+2588T4−37552T6+401926T8−37552p2T10+2588p4T12−80p6T14+p8T16 1 - 80 T^{2} + 2588 T^{4} - 37552 T^{6} + 401926 T^{8} - 37552 p^{2} T^{10} + 2588 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16}
29 (1+2T+73T2+124T3+3004T4+124pT5+73p2T6+2p3T7+p4T8)2 ( 1 + 2 T + 73 T^{2} + 124 T^{3} + 3004 T^{4} + 124 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
31 (1−2T+9T2+14T3+636T4+14pT5+9p2T6−2p3T7+p4T8)2 ( 1 - 2 T + 9 T^{2} + 14 T^{3} + 636 T^{4} + 14 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
37 (1+6T+77T2+8T3+1572T4+8pT5+77p2T6+6p3T7+p4T8)2 ( 1 + 6 T + 77 T^{2} + 8 T^{3} + 1572 T^{4} + 8 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
41 (1+8T+152T2+904T3+9166T4+904pT5+152p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 152 T^{2} + 904 T^{3} + 9166 T^{4} + 904 p T^{5} + 152 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 1−184T2+16700T4−1008840T6+47686758T8−1008840p2T10+16700p4T12−184p6T14+p8T16 1 - 184 T^{2} + 16700 T^{4} - 1008840 T^{6} + 47686758 T^{8} - 1008840 p^{2} T^{10} + 16700 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16}
47 1−120T2+8828T4−511176T6+26922502T8−511176p2T10+8828p4T12−120p6T14+p8T16 1 - 120 T^{2} + 8828 T^{4} - 511176 T^{6} + 26922502 T^{8} - 511176 p^{2} T^{10} + 8828 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16}
53 1−366T2+60961T4−6049838T6+392003636T8−6049838p2T10+60961p4T12−366p6T14+p8T16 1 - 366 T^{2} + 60961 T^{4} - 6049838 T^{6} + 392003636 T^{8} - 6049838 p^{2} T^{10} + 60961 p^{4} T^{12} - 366 p^{6} T^{14} + p^{8} T^{16}
59 1−284T2+41876T4−4064260T6+281102518T8−4064260p2T10+41876p4T12−284p6T14+p8T16 1 - 284 T^{2} + 41876 T^{4} - 4064260 T^{6} + 281102518 T^{8} - 4064260 p^{2} T^{10} + 41876 p^{4} T^{12} - 284 p^{6} T^{14} + p^{8} T^{16}
61 1−178T2+16233T4−1367798T6+99891260T8−1367798p2T10+16233p4T12−178p6T14+p8T16 1 - 178 T^{2} + 16233 T^{4} - 1367798 T^{6} + 99891260 T^{8} - 1367798 p^{2} T^{10} + 16233 p^{4} T^{12} - 178 p^{6} T^{14} + p^{8} T^{16}
67 (1−6T+92T2−270T3+7174T4−270pT5+92p2T6−6p3T7+p4T8)2 ( 1 - 6 T + 92 T^{2} - 270 T^{3} + 7174 T^{4} - 270 p T^{5} + 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
71 1−398T2+73505T4−8497986T6+699860940T8−8497986p2T10+73505p4T12−398p6T14+p8T16 1 - 398 T^{2} + 73505 T^{4} - 8497986 T^{6} + 699860940 T^{8} - 8497986 p^{2} T^{10} + 73505 p^{4} T^{12} - 398 p^{6} T^{14} + p^{8} T^{16}
73 1−204T2+21444T4−1995828T6+165946934T8−1995828p2T10+21444p4T12−204p6T14+p8T16 1 - 204 T^{2} + 21444 T^{4} - 1995828 T^{6} + 165946934 T^{8} - 1995828 p^{2} T^{10} + 21444 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16}
79 1−4pT2+50260T4−5841924T6+529552214T8−5841924p2T10+50260p4T12−4p7T14+p8T16 1 - 4 p T^{2} + 50260 T^{4} - 5841924 T^{6} + 529552214 T^{8} - 5841924 p^{2} T^{10} + 50260 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}
83 (1−36T+724T2−9924T3+103462T4−9924pT5+724p2T6−36p3T7+p4T8)2 ( 1 - 36 T + 724 T^{2} - 9924 T^{3} + 103462 T^{4} - 9924 p T^{5} + 724 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2}
89 1−450T2+88625T4−10694546T6+1008754980T8−10694546p2T10+88625p4T12−450p6T14+p8T16 1 - 450 T^{2} + 88625 T^{4} - 10694546 T^{6} + 1008754980 T^{8} - 10694546 p^{2} T^{10} + 88625 p^{4} T^{12} - 450 p^{6} T^{14} + p^{8} T^{16}
97 (1+4T+172T2+732T3+25958T4+732pT5+172p2T6+4p3T7+p4T8)2 ( 1 + 4 T + 172 T^{2} + 732 T^{3} + 25958 T^{4} + 732 p T^{5} + 172 p^{2} T^{6} + 4 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

−3.63774262628700070316077480001, −3.45011738477178693693207448465, −3.35309982136982853390631045208, −3.31198775104654406891394977185, −3.27571456929968347869723990489, −3.19573958135364787287280103492, −3.06645209727711529753815696749, −2.98958180338778428871583008760, −2.76113577361030613726203250699, −2.42406490268166475222079481014, −2.27773480251639995818024650769, −2.24384590154903788338620633738, −2.15785693020796979252558298968, −2.07350992954652736834564294631, −2.01479665726920595455387385774, −1.90666043504541226691131731123, −1.78459550467572788329966557681, −1.32723689228536405378999956413, −1.24244507319561031508824570752, −1.04481745562654336350814567837, −0.876213955070272641785123011344, −0.78276280535055089158301410891, −0.54339682428973169266482312306, −0.49362768118819703381522321162, −0.14215217647355781052957450520, 0.14215217647355781052957450520, 0.49362768118819703381522321162, 0.54339682428973169266482312306, 0.78276280535055089158301410891, 0.876213955070272641785123011344, 1.04481745562654336350814567837, 1.24244507319561031508824570752, 1.32723689228536405378999956413, 1.78459550467572788329966557681, 1.90666043504541226691131731123, 2.01479665726920595455387385774, 2.07350992954652736834564294631, 2.15785693020796979252558298968, 2.24384590154903788338620633738, 2.27773480251639995818024650769, 2.42406490268166475222079481014, 2.76113577361030613726203250699, 2.98958180338778428871583008760, 3.06645209727711529753815696749, 3.19573958135364787287280103492, 3.27571456929968347869723990489, 3.31198775104654406891394977185, 3.35309982136982853390631045208, 3.45011738477178693693207448465, 3.63774262628700070316077480001

Graph of the ZZ-function along the critical line

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