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Properties

Label 16-3332e8-1.1-c0e8-0-6
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $58.4651$
Root an. cond. $1.28952$
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  + 16-s + 2·25-s − 4·37-s − 8·41-s + 4·53-s + 4·61-s + 8·101-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 16-s + 2·25-s − 4·37-s − 8·41-s + 4·53-s + 4·61-s + 8·101-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(58.4651\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.677833456\)
\(L(\frac12)\) \(\approx\) \(1.677833456\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.67041184292760459942877887781, −3.62955722604908126287703879846, −3.53494915371445483046303017657, −3.41771687755488308128885370052, −3.34899124692051940482125257461, −3.30949861999462570812819989718, −3.28046304674850454497978475998, −3.03258338465119621035031858123, −2.96917741610359589758718964647, −2.76597752983309755168459772768, −2.46550938726616524475874508777, −2.43011682657594202268918753013, −2.35909583238435316373018166071, −2.11276847089133480192130302580, −2.04036288934781870613153549600, −1.87578948984808099513278947038, −1.86137447689934077097655311697, −1.81659003137354326186323287432, −1.72835739662762071265245408365, −1.17118602317589879789733212688, −1.13907200535183769131176040800, −1.05740136694526897986230506952, −1.05328748996031345716370664575, −0.58539248629331278947778383125, −0.38077843168179938311252986372, 0.38077843168179938311252986372, 0.58539248629331278947778383125, 1.05328748996031345716370664575, 1.05740136694526897986230506952, 1.13907200535183769131176040800, 1.17118602317589879789733212688, 1.72835739662762071265245408365, 1.81659003137354326186323287432, 1.86137447689934077097655311697, 1.87578948984808099513278947038, 2.04036288934781870613153549600, 2.11276847089133480192130302580, 2.35909583238435316373018166071, 2.43011682657594202268918753013, 2.46550938726616524475874508777, 2.76597752983309755168459772768, 2.96917741610359589758718964647, 3.03258338465119621035031858123, 3.28046304674850454497978475998, 3.30949861999462570812819989718, 3.34899124692051940482125257461, 3.41771687755488308128885370052, 3.53494915371445483046303017657, 3.62955722604908126287703879846, 3.67041184292760459942877887781

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

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