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Properties

Label 16-624e8-1.1-c0e8-0-0
Degree 1616
Conductor 2.299×10222.299\times 10^{22}
Sign 11
Analytic cond. 8.84573×10−58.84573\times 10^{-5}
Root an. cond. 0.5580470.558047
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  − 8·43-s − 8·49-s − 2·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 8·43-s − 8·49-s − 2·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

Λ(s)=((232⋅38⋅138)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((232⋅38⋅138)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 232⋅38⋅1382^{32} \cdot 3^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 8.84573×10−58.84573\times 10^{-5}
Root analytic conductor: 0.5580470.558047
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 232⋅38⋅138, ( :[0]8), 1)(16,\ 2^{32} \cdot 3^{8} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.27276024420.2727602442
L(12)L(\frac12) ≈\approx 0.27276024420.2727602442
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+T8 1 + T^{8}
3 (1+T4)2 ( 1 + T^{4} )^{2}
13 (1+T4)2 ( 1 + T^{4} )^{2}
good5 (1+T8)2 ( 1 + T^{8} )^{2}
7 (1+T2)8 ( 1 + T^{2} )^{8}
11 (1+T8)2 ( 1 + T^{8} )^{2}
17 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
19 (1+T4)4 ( 1 + T^{4} )^{4}
23 (1+T2)8 ( 1 + T^{2} )^{8}
29 (1+T4)4 ( 1 + T^{4} )^{4}
31 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
37 (1+T4)4 ( 1 + T^{4} )^{4}
41 (1+T8)2 ( 1 + T^{8} )^{2}
43 (1+T)8(1+T2)4 ( 1 + T )^{8}( 1 + T^{2} )^{4}
47 (1+T8)2 ( 1 + T^{8} )^{2}
53 (1+T4)4 ( 1 + T^{4} )^{4}
59 (1+T8)2 ( 1 + T^{8} )^{2}
61 (1+T4)4 ( 1 + T^{4} )^{4}
67 (1+T4)4 ( 1 + T^{4} )^{4}
71 (1+T8)2 ( 1 + T^{8} )^{2}
73 (1+T2)8 ( 1 + T^{2} )^{8}
79 (1+T4)4 ( 1 + T^{4} )^{4}
83 (1+T8)2 ( 1 + T^{8} )^{2}
89 (1+T8)2 ( 1 + T^{8} )^{2}
97 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
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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

−4.95453201455515120322607667958, −4.83691777022271644997221636205, −4.69170158630456900308473115441, −4.54734357132608278599234076129, −4.27922511120601303739164792078, −4.27391684529640215235864146333, −4.15941798208239544213654513720, −3.96959940041395700049470072592, −3.49626980091899211900167495765, −3.47568573785165170670278817830, −3.42137822706251170721136390781, −3.35600686624839614393160237109, −3.32708304348950427058141625128, −3.18891818677673892125455825707, −2.85007558694573951370941669280, −2.74691665918881862604545528709, −2.56558107265464721939571014085, −2.53530285833621762441390648403, −1.87201397448150074188789799761, −1.74446295348748728551273560509, −1.70273815481864581087862194144, −1.68807888608735225006993147654, −1.61868251366773896733852901583, −1.39352023955886229394312368819, −0.56799868889114961379323554390, 0.56799868889114961379323554390, 1.39352023955886229394312368819, 1.61868251366773896733852901583, 1.68807888608735225006993147654, 1.70273815481864581087862194144, 1.74446295348748728551273560509, 1.87201397448150074188789799761, 2.53530285833621762441390648403, 2.56558107265464721939571014085, 2.74691665918881862604545528709, 2.85007558694573951370941669280, 3.18891818677673892125455825707, 3.32708304348950427058141625128, 3.35600686624839614393160237109, 3.42137822706251170721136390781, 3.47568573785165170670278817830, 3.49626980091899211900167495765, 3.96959940041395700049470072592, 4.15941798208239544213654513720, 4.27391684529640215235864146333, 4.27922511120601303739164792078, 4.54734357132608278599234076129, 4.69170158630456900308473115441, 4.83691777022271644997221636205, 4.95453201455515120322607667958

Graph of the ZZ-function along the critical line

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