# 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 of factors

## 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}$$
show less
$$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

−4.28413145366950377734966843195, −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.