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Properties

Label 16-329e8-1.1-c0e8-0-0
Degree 1616
Conductor 1.373×10201.373\times 10^{20}
Sign 11
Analytic cond. 5.28231×10−75.28231\times 10^{-7}
Root an. cond. 0.4052060.405206
Motivic weight 00
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-s + 2·3-s − 2·6-s + 7-s − 8-s + 3·9-s − 14-s + 16-s − 17-s − 3·18-s + 2·21-s − 2·24-s − 4·25-s + 2·27-s + 32-s + 34-s − 37-s − 2·42-s − 4·47-s + 2·48-s + 49-s + 4·50-s − 2·51-s − 53-s − 2·54-s − 56-s − 59-s + ⋯
L(s)  = 1  − 2-s + 2·3-s − 2·6-s + 7-s − 8-s + 3·9-s − 14-s + 16-s − 17-s − 3·18-s + 2·21-s − 2·24-s − 4·25-s + 2·27-s + 32-s + 34-s − 37-s − 2·42-s − 4·47-s + 2·48-s + 49-s + 4·50-s − 2·51-s − 53-s − 2·54-s − 56-s − 59-s + ⋯

Functional equation

Λ(s)=((78⋅478)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((78⋅478)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 78⋅4787^{8} \cdot 47^{8}
Sign: 11
Analytic conductor: 5.28231×10−75.28231\times 10^{-7}
Root analytic conductor: 0.4052060.405206
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 78⋅478, ( :[0]8), 1)(16,\ 7^{8} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.15274709710.1527470971
L(12)L(\frac12) ≈\approx 0.15274709710.1527470971
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)
bad7 1−T+T3−T4+T5−T7+T8 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8}
47 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
good2 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
3 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
5 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
11 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
13 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
17 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
19 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
23 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
29 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
31 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
37 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
41 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
43 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
53 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
59 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
61 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
67 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
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)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
79 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
83 (1+T+T2)8 ( 1 + T + T^{2} )^{8}
89 (1−T+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
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

−5.47907303025714143626134814770, −5.18118346552035789536329621453, −5.13171704548076107549426400459, −5.06753779487272236067716293054, −4.86023436445515047227807239936, −4.76386455813186233448959683750, −4.31942202782266471653121033957, −4.31284991009843045661816457700, −4.30098233821970231797049191624, −4.07457665983559261799411358079, −3.86654237710398461413584328865, −3.85318803289625649839914512541, −3.64692453493006510428542000412, −3.35300331510020270218540063404, −3.30804352399552803289161103522, −2.98684283668355827670913848217, −2.90183104402826162231304474382, −2.86233327996632195942287932181, −2.41612396739851796074505688589, −2.26449363185006760463762289466, −2.11133047432001115468590785707, −1.80868600520281066469065676046, −1.68645756168233095062941011887, −1.48246323250519140429948159420, −1.47036536131553938298971534235, 1.47036536131553938298971534235, 1.48246323250519140429948159420, 1.68645756168233095062941011887, 1.80868600520281066469065676046, 2.11133047432001115468590785707, 2.26449363185006760463762289466, 2.41612396739851796074505688589, 2.86233327996632195942287932181, 2.90183104402826162231304474382, 2.98684283668355827670913848217, 3.30804352399552803289161103522, 3.35300331510020270218540063404, 3.64692453493006510428542000412, 3.85318803289625649839914512541, 3.86654237710398461413584328865, 4.07457665983559261799411358079, 4.30098233821970231797049191624, 4.31284991009843045661816457700, 4.31942202782266471653121033957, 4.76386455813186233448959683750, 4.86023436445515047227807239936, 5.06753779487272236067716293054, 5.13171704548076107549426400459, 5.18118346552035789536329621453, 5.47907303025714143626134814770

Graph of the ZZ-function along the critical line

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