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Properties

Label 16-465e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.186\times 10^{21}$
Sign $1$
Analytic cond. $8.41163\times 10^{-6}$
Root an. cond. $0.481731$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 3·4-s − 4·5-s + 2·6-s + 2·8-s + 9-s − 8·10-s + 3·12-s − 4·15-s + 16-s − 17-s + 2·18-s − 6·19-s − 12·20-s + 6·23-s + 2·24-s + 6·25-s − 8·30-s − 2·31-s − 2·32-s − 2·34-s + 3·36-s − 12·38-s − 8·40-s − 4·45-s + 12·46-s + ⋯
L(s)  = 1  + 2·2-s + 3-s + 3·4-s − 4·5-s + 2·6-s + 2·8-s + 9-s − 8·10-s + 3·12-s − 4·15-s + 16-s − 17-s + 2·18-s − 6·19-s − 12·20-s + 6·23-s + 2·24-s + 6·25-s − 8·30-s − 2·31-s − 2·32-s − 2·34-s + 3·36-s − 12·38-s − 8·40-s − 4·45-s + 12·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(8.41163\times 10^{-6}\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4819314101\)
\(L(\frac12)\) \(\approx\) \(0.4819314101\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
5 \( ( 1 + T + T^{2} )^{4} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
19 \( ( 1 + T + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.09505852012124105691851996524, −4.90442813395917164895037239582, −4.64018499297231635875271492843, −4.45186882938014062509276643625, −4.44586980775492035884104680437, −4.43150606191911858825752555521, −4.23141662749007369210737394177, −4.03076614863049817693286838980, −3.98092773121881294086306835642, −3.94066321158446791085238029042, −3.86350197024384300628364228715, −3.47246675472654939217306315104, −3.46579053368725549075210533157, −3.20449759616214397780605774626, −3.16599391802978997064997536895, −3.06962902934039014613600106614, −3.02517143125482888083573226766, −2.46879601323785015930959832518, −2.32113414567593505922990885568, −2.28910620691738664547231703500, −2.25741699819250812528410686858, −1.91777998220283602764779490013, −1.83174706930767172794147089292, −1.21024690129198319322793724669, −0.819544924012247194107594640238, 0.819544924012247194107594640238, 1.21024690129198319322793724669, 1.83174706930767172794147089292, 1.91777998220283602764779490013, 2.25741699819250812528410686858, 2.28910620691738664547231703500, 2.32113414567593505922990885568, 2.46879601323785015930959832518, 3.02517143125482888083573226766, 3.06962902934039014613600106614, 3.16599391802978997064997536895, 3.20449759616214397780605774626, 3.46579053368725549075210533157, 3.47246675472654939217306315104, 3.86350197024384300628364228715, 3.94066321158446791085238029042, 3.98092773121881294086306835642, 4.03076614863049817693286838980, 4.23141662749007369210737394177, 4.43150606191911858825752555521, 4.44586980775492035884104680437, 4.45186882938014062509276643625, 4.64018499297231635875271492843, 4.90442813395917164895037239582, 5.09505852012124105691851996524

Graph of the $Z$-function along the critical line

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