↑

Properties

Label 16-3024e8-1.1-c0e8-0-1
Degree $16$
Conductor $6.993\times 10^{27}$
Sign $1$
Analytic cond. $26.9098$
Root an. cond. $1.22848$
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  + 16-s + 4·19-s + 8·31-s + 8·43-s − 4·49-s − 4·67-s − 4·79-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 + ⋯
L(s)  = 1  + 16-s + 4·19-s + 8·31-s + 8·43-s − 4·49-s − 4·67-s − 4·79-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(26.9098\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.630975946\)
\(L(\frac12)\) \(\approx\) \(4.630975946\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T^{4} + T^{8} )^{2} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 - T + T^{2} )^{8} \)
37 \( ( 1 - T^{4} + T^{8} )^{2} \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
61 \( ( 1 - T^{4} + T^{8} )^{2} \)
67 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T^{2} )^{8} \)
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

−3.75171179099940555968076289632, −3.69638297168484565334584674892, −3.69219961305865987664064297977, −3.65453594826459800850291363712, −3.03670903627529021534646618706, −3.03518764318064512441326084884, −2.99690401581498661889506370458, −2.99303921240526502117730731222, −2.97500778556655159749187244935, −2.83324131999761806573908414646, −2.73660975480310330487908834705, −2.66520510591178828963953080102, −2.52720668739560390121540969417, −2.40752488645712053058207063247, −2.32610254600102098701600992812, −1.83372061526956496553946390475, −1.66812580235461954340306089309, −1.64217398508412690628041009443, −1.50706660547553309075909611246, −1.41047235965075059402506544927, −1.10277731143920598696381122967, −0.950765345401267503076895359611, −0.924461253475204145556098802084, −0.877223318839565187864379131796, −0.66184400862286512551204337731, 0.66184400862286512551204337731, 0.877223318839565187864379131796, 0.924461253475204145556098802084, 0.950765345401267503076895359611, 1.10277731143920598696381122967, 1.41047235965075059402506544927, 1.50706660547553309075909611246, 1.64217398508412690628041009443, 1.66812580235461954340306089309, 1.83372061526956496553946390475, 2.32610254600102098701600992812, 2.40752488645712053058207063247, 2.52720668739560390121540969417, 2.66520510591178828963953080102, 2.73660975480310330487908834705, 2.83324131999761806573908414646, 2.97500778556655159749187244935, 2.99303921240526502117730731222, 2.99690401581498661889506370458, 3.03518764318064512441326084884, 3.03670903627529021534646618706, 3.65453594826459800850291363712, 3.69219961305865987664064297977, 3.69638297168484565334584674892, 3.75171179099940555968076289632

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

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