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Properties

Label 16-1008e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.066\times 10^{24}$
Sign $1$
Analytic cond. $0.00410148$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 8·13-s + 16-s + 6·49-s − 4·61-s − 32·91-s + 8·97-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 8·208-s + 211-s + ⋯
L(s)  = 1  − 4·7-s + 8·13-s + 16-s + 6·49-s − 4·61-s − 32·91-s + 8·97-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 8·208-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.00410148\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7888410615\)
\(L(\frac12)\) \(\approx\) \(0.7888410615\)
\(L(1)\) 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 \( ( 1 + T + T^{2} )^{4} \)
good5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
17 \( ( 1 - T^{2} + T^{4} )^{4} \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
37 \( ( 1 - T^{4} + T^{8} )^{2} \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 + T^{4} )^{4} \)
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 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T^{4} + T^{8} )^{2} \)
97 \( ( 1 - T + T^{2} )^{8} \)
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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

−4.48669013845557381324866677537, −4.24526414141190827461049336074, −4.20282713880007494650671736155, −4.00655357663277821578341180719, −3.77052539717295612858146967316, −3.71478167488610124515557801803, −3.71008619255601691977672132216, −3.54270627502416413623786514910, −3.44869801082743206655978634160, −3.40082005660433289512882771635, −3.28696684098421995474206715054, −3.20975294635891894482731796909, −3.07913044444080022296992620655, −2.88818597623519621398257230510, −2.75896890021932332679562068305, −2.56618236682001401198226931814, −2.38746767481844415820798082049, −1.96349569250233831299398310348, −1.78492534386597582453196419161, −1.73385282925096996140256921178, −1.57687872548948320736883100339, −1.34773672530009059790906529188, −0.955069714973560626517743716396, −0.946475793828449386761839104308, −0.916555695665794486249253442370, 0.916555695665794486249253442370, 0.946475793828449386761839104308, 0.955069714973560626517743716396, 1.34773672530009059790906529188, 1.57687872548948320736883100339, 1.73385282925096996140256921178, 1.78492534386597582453196419161, 1.96349569250233831299398310348, 2.38746767481844415820798082049, 2.56618236682001401198226931814, 2.75896890021932332679562068305, 2.88818597623519621398257230510, 3.07913044444080022296992620655, 3.20975294635891894482731796909, 3.28696684098421995474206715054, 3.40082005660433289512882771635, 3.44869801082743206655978634160, 3.54270627502416413623786514910, 3.71008619255601691977672132216, 3.71478167488610124515557801803, 3.77052539717295612858146967316, 4.00655357663277821578341180719, 4.20282713880007494650671736155, 4.24526414141190827461049336074, 4.48669013845557381324866677537

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

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