Properties

Label 16-1620e8-1.1-c0e8-0-0
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $0.182548$
Root an. cond. $0.899158$
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  − 8·13-s + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 8·13-s + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.182548\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1860210767\)
\(L(\frac12)\) \(\approx\) \(0.1860210767\)
\(L(1)\) 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 \( 1 \)
5 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
41 \( ( 1 - T^{4} + T^{8} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T^{4} + T^{8} )^{2} \)
97 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
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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

−4.28413145366950377734966843195, −4.23533263402240901755350994897, −4.09112577556449286261716250049, −3.96194431360034616461448886154, −3.61472686744484970078330187291, −3.52442965882191482583744149730, −3.51370806710728590956459472029, −3.15791385918476721056430636178, −2.96442023412964206608957295311, −2.93127897965490121460443675320, −2.88862077082545931475789721635, −2.81585811906944623297504170800, −2.63780158461373181775080698332, −2.62495250444300182537504969738, −2.44074904524439490058349553075, −2.23196682838475065824164388948, −2.22045349369652357799691041359, −1.94105871708479458431107976979, −1.76460930281198264879282382809, −1.71537300918316557586658568835, −1.66423372131465339649807052837, −1.06452112433471316788125502474, −0.935509401911315353931934281498, −0.816665320780349375061193051129, −0.22036881706556110741650052589, 0.22036881706556110741650052589, 0.816665320780349375061193051129, 0.935509401911315353931934281498, 1.06452112433471316788125502474, 1.66423372131465339649807052837, 1.71537300918316557586658568835, 1.76460930281198264879282382809, 1.94105871708479458431107976979, 2.22045349369652357799691041359, 2.23196682838475065824164388948, 2.44074904524439490058349553075, 2.62495250444300182537504969738, 2.63780158461373181775080698332, 2.81585811906944623297504170800, 2.88862077082545931475789721635, 2.93127897965490121460443675320, 2.96442023412964206608957295311, 3.15791385918476721056430636178, 3.51370806710728590956459472029, 3.52442965882191482583744149730, 3.61472686744484970078330187291, 3.96194431360034616461448886154, 4.09112577556449286261716250049, 4.23533263402240901755350994897, 4.28413145366950377734966843195

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

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