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Properties

Label 16-968e8-1.1-c0e8-0-0
Degree 1616
Conductor 7.709×10237.709\times 10^{23}
Sign 11
Analytic cond. 0.002966600.00296660
Root an. cond. 0.6950500.695050
Motivic weight 00
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  + 2·3-s − 2·5-s + 3·9-s − 4·15-s − 8·23-s + 3·25-s + 2·27-s − 2·31-s − 2·37-s − 6·45-s − 2·59-s + 8·67-s − 16·69-s + 2·71-s + 6·75-s + 81-s − 8·89-s − 4·93-s + 2·97-s − 4·111-s − 2·113-s + 16·115-s − 2·125-s + 127-s + 131-s − 4·135-s + 137-s + ⋯
L(s)  = 1  + 2·3-s − 2·5-s + 3·9-s − 4·15-s − 8·23-s + 3·25-s + 2·27-s − 2·31-s − 2·37-s − 6·45-s − 2·59-s + 8·67-s − 16·69-s + 2·71-s + 6·75-s + 81-s − 8·89-s − 4·93-s + 2·97-s − 4·111-s − 2·113-s + 16·115-s − 2·125-s + 127-s + 131-s − 4·135-s + 137-s + ⋯

Functional equation

Λ(s)=((224⋅1116)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((224⋅1116)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 224⋅11162^{24} \cdot 11^{16}
Sign: 11
Analytic conductor: 0.002966600.00296660
Root analytic conductor: 0.6950500.695050
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 224⋅1116, ( :[0]8), 1)(16,\ 2^{24} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.35867209610.3586720961
L(12)L(\frac12) ≈\approx 0.35867209610.3586720961
L(1)L(1) 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
11 1 1
good3 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
5 (1+T−T3−T4−T5+T7+T8)2 ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}
7 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
13 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
17 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
19 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
23 (1+T+T2)8 ( 1 + T + T^{2} )^{8}
29 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
31 (1+T−T3−T4−T5+T7+T8)2 ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}
37 (1+T−T3−T4−T5+T7+T8)2 ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}
41 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
43 (1+T4)4 ( 1 + T^{4} )^{4}
47 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
53 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
59 (1+T−T3−T4−T5+T7+T8)2 ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}
61 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
67 (1−T+T2)8 ( 1 - T + T^{2} )^{8}
71 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
73 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
79 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
83 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
89 (1+T+T2)8 ( 1 + T + T^{2} )^{8}
97 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + 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

−4.32801509692024875467776243308, −4.21668534274142585456052731911, −4.13712600598296971717671369452, −4.13077820116991630063198439689, −3.84433360353964124388669971458, −3.80840099383446542553098060016, −3.78567196601666207751237357000, −3.61729960179301366850380031045, −3.58418886820362785195632855123, −3.57700461717680160225671419087, −3.56303110613607760298212309982, −3.06333174787931476590933791131, −2.79575549184256168646708456511, −2.77923755223142338495361035367, −2.59608908758388173208369220750, −2.57990533760632707977610532407, −2.21110414080452641908912603621, −2.07539870427375014707055299856, −1.95909544567123710920095841052, −1.80045032489689858169181402559, −1.79232013696017646061738422107, −1.72894665450767901338040088902, −1.26294153761128970126630603489, −1.04052006312900831254152696419, −0.34730094615671320267926251175, 0.34730094615671320267926251175, 1.04052006312900831254152696419, 1.26294153761128970126630603489, 1.72894665450767901338040088902, 1.79232013696017646061738422107, 1.80045032489689858169181402559, 1.95909544567123710920095841052, 2.07539870427375014707055299856, 2.21110414080452641908912603621, 2.57990533760632707977610532407, 2.59608908758388173208369220750, 2.77923755223142338495361035367, 2.79575549184256168646708456511, 3.06333174787931476590933791131, 3.56303110613607760298212309982, 3.57700461717680160225671419087, 3.58418886820362785195632855123, 3.61729960179301366850380031045, 3.78567196601666207751237357000, 3.80840099383446542553098060016, 3.84433360353964124388669971458, 4.13077820116991630063198439689, 4.13712600598296971717671369452, 4.21668534274142585456052731911, 4.32801509692024875467776243308

Graph of the ZZ-function along the critical line

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