↑

Properties

Label 16-1148e8-1.1-c0e8-0-2
Degree $16$
Conductor $3.017\times 10^{24}$
Sign $1$
Analytic cond. $0.0116089$
Root an. cond. $0.756919$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·4-s − 15·16-s − 2·25-s − 24·32-s + 8·41-s − 8·50-s − 4·61-s − 6·64-s − 81-s + 32·82-s − 12·100-s − 16·122-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·162-s + 163-s + 48·164-s + 167-s + 8·169-s + 173-s + ⋯
L(s)  = 1  + 4·2-s + 6·4-s − 15·16-s − 2·25-s − 24·32-s + 8·41-s − 8·50-s − 4·61-s − 6·64-s − 81-s + 32·82-s − 12·100-s − 16·122-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·162-s + 163-s + 48·164-s + 167-s + 8·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{8} \cdot 41^{8}\)
Sign: $1$
Analytic conductor: \(0.0116089\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{8} \cdot 41^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.094653021\)
\(L(\frac12)\) \(\approx\) \(3.094653021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{4} \)
7 \( 1 - T^{4} + T^{8} \)
41 \( ( 1 - T )^{8} \)
good3 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
5 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
23 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T^{4} + T^{8} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} )^{4} \)
79 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.42470825690778690460718283740, −4.31470339002634743473703539665, −4.24946721618541658633344012891, −4.03576882114875450705771651254, −3.99918958122409024678437666689, −3.89126020310432153426250619893, −3.80321237972504559655340213589, −3.79541892039467770517004050140, −3.50189059843053781102807040703, −3.26727301950420158085170848470, −3.15870938995719671389297045974, −2.94684129174437532275751933013, −2.91138898923709232084090716102, −2.82038984406950722036720502042, −2.81962520485208165693112483111, −2.63367365765967059065735941983, −2.58665123259185702685698889829, −2.10764208365349026658219098080, −2.03759153083777961913956844131, −1.87003989497706340214766953971, −1.81937357046570986560164506648, −1.59438215829365326960239542714, −0.866174618577696087774110532668, −0.832651951057402292297510188007, −0.76277230857141769827968820270, 0.76277230857141769827968820270, 0.832651951057402292297510188007, 0.866174618577696087774110532668, 1.59438215829365326960239542714, 1.81937357046570986560164506648, 1.87003989497706340214766953971, 2.03759153083777961913956844131, 2.10764208365349026658219098080, 2.58665123259185702685698889829, 2.63367365765967059065735941983, 2.81962520485208165693112483111, 2.82038984406950722036720502042, 2.91138898923709232084090716102, 2.94684129174437532275751933013, 3.15870938995719671389297045974, 3.26727301950420158085170848470, 3.50189059843053781102807040703, 3.79541892039467770517004050140, 3.80321237972504559655340213589, 3.89126020310432153426250619893, 3.99918958122409024678437666689, 4.03576882114875450705771651254, 4.24946721618541658633344012891, 4.31470339002634743473703539665, 4.42470825690778690460718283740

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

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