Properties

Degree 16
Conductor $ 2^{40} \cdot 3^{24} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 24·13-s − 16·25-s − 8·37-s − 4·49-s + 16·61-s + 40·73-s − 48·97-s − 16·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 232·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 6.65·13-s − 3.19·25-s − 1.31·37-s − 4/7·49-s + 2.04·61-s + 4.68·73-s − 4.87·97-s − 1.53·109-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 17.8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 3^{24} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.3523351266$
$L(\frac12)$  $\approx$  $0.3523351266$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 16 T^{2} + 21 p T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 88 T^{2} + 2991 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 2 T^{2} - 429 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 8 T^{2} + 2055 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 148 T^{2} + 9066 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 112 T^{2} + 6786 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 100 T^{2} + 7686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 184 T^{2} + 15234 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 116 T^{2} + 12042 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 128 T^{2} + 8103 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 164 T^{2} + 13914 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 40 T^{2} - 1374 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 352 T^{2} + 46815 T^{4} - 352 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.13973349923110398369655981234, −3.04040113124639910722115681060, −2.97921419108854785323029544242, −2.92862919404095821200058885417, −2.77126538799763323180015285737, −2.70822838627461790142793633777, −2.68169178109197688005205847913, −2.44800274358054743428837571586, −2.38337746568990344065629523210, −2.25268116770506600838995982866, −2.24672272708011926796611399404, −2.02797123380426936886867699404, −1.87873516534730473316562516984, −1.87367349928948154386784726585, −1.82855553907130628820850649175, −1.64504207121137892434809411434, −1.64136816663485923068009508528, −1.12688807482298077541878111338, −1.03518939179099225735620068217, −0.992094567855113106685363102484, −0.855021973796694489965699293567, −0.47875312469303823251492868929, −0.24150242015579209210053678799, −0.22948232768961043615105816017, −0.13759393097661429038370973535, 0.13759393097661429038370973535, 0.22948232768961043615105816017, 0.24150242015579209210053678799, 0.47875312469303823251492868929, 0.855021973796694489965699293567, 0.992094567855113106685363102484, 1.03518939179099225735620068217, 1.12688807482298077541878111338, 1.64136816663485923068009508528, 1.64504207121137892434809411434, 1.82855553907130628820850649175, 1.87367349928948154386784726585, 1.87873516534730473316562516984, 2.02797123380426936886867699404, 2.24672272708011926796611399404, 2.25268116770506600838995982866, 2.38337746568990344065629523210, 2.44800274358054743428837571586, 2.68169178109197688005205847913, 2.70822838627461790142793633777, 2.77126538799763323180015285737, 2.92862919404095821200058885417, 2.97921419108854785323029544242, 3.04040113124639910722115681060, 3.13973349923110398369655981234

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

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