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  + 8·7-s + 24·25-s + 56·31-s + 36·49-s − 24·73-s − 8·79-s − 16·97-s + 24·103-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 192·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 3.02·7-s + 24/5·25-s + 10.0·31-s + 36/7·49-s − 2.80·73-s − 0.900·79-s − 1.62·97-s + 2.36·103-s − 0.727·121-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 + 3.07·169-s + 0.0760·173-s + 14.5·175-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 + ⋯

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  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $132.5009827$
$L(\frac12)$  $\approx$  $132.5009827$
$L(\frac{3}{2})$   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 \( ( 1 - T )^{8} \)
good5 \( ( 1 - 12 T^{2} + 71 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 4 T^{2} + 111 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 20 T^{2} + 378 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 14 T^{2} - 189 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 84 T^{2} + 2807 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 7 T + p T^{2} )^{8} \)
37 \( ( 1 - 110 T^{2} + 5523 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 12 T^{2} + 2663 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 92 T^{2} + 4314 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - p T^{2} )^{8} \)
59 \( ( 1 - 24 T^{2} + 3266 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 52 T^{2} - 522 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 20 T^{2} + 4218 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 236 T^{2} + 23871 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 120 T^{2} + 13538 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 204 T^{2} + 25511 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
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

−3.18046102589403938302221360822, −3.16567433713563689814407166149, −2.93373838658257627966009446600, −2.89272636277178192599650976837, −2.86609725201639784332249962472, −2.82424016999826177957495345324, −2.72682214030002471775119679719, −2.58596582219321154984449818660, −2.39193587385497847038686418184, −2.20619947455441116960769008433, −2.18456319277669643861572980042, −2.06173850842122395883349815189, −2.05904705587709057306272508772, −1.63483635829392018673831712996, −1.54765269465188050733320215034, −1.50629415391339233863319085576, −1.36825121604899471126493705249, −1.15590604749063937512279275809, −1.12485555268340234170082299514, −1.04841944900664709962520175307, −0.942717416648375686788223624202, −0.69525190954627173304912226029, −0.61590984346459155109150949118, −0.44299111205426556689427827113, −0.43441712241863975160851200579, 0.43441712241863975160851200579, 0.44299111205426556689427827113, 0.61590984346459155109150949118, 0.69525190954627173304912226029, 0.942717416648375686788223624202, 1.04841944900664709962520175307, 1.12485555268340234170082299514, 1.15590604749063937512279275809, 1.36825121604899471126493705249, 1.50629415391339233863319085576, 1.54765269465188050733320215034, 1.63483635829392018673831712996, 2.05904705587709057306272508772, 2.06173850842122395883349815189, 2.18456319277669643861572980042, 2.20619947455441116960769008433, 2.39193587385497847038686418184, 2.58596582219321154984449818660, 2.72682214030002471775119679719, 2.82424016999826177957495345324, 2.86609725201639784332249962472, 2.89272636277178192599650976837, 2.93373838658257627966009446600, 3.16567433713563689814407166149, 3.18046102589403938302221360822

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

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