Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 31·16-s − 72·23-s − 232·43-s + 24·53-s − 472·67-s + 744·107-s + 424·109-s + 412·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 1.93·16-s − 3.13·23-s − 5.39·43-s + 0.452·53-s − 7.04·67-s + 6.95·107-s + 3.88·109-s + 3.40·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{40} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(4.48518\)
\(L(\frac12)\)  \(\approx\)  \(4.48518\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 - 31 T^{4} + p^{8} T^{8} \)
7 \( ( 1 + p^{4} T^{4} )^{2} \)
good3 \( ( 1 + p^{8} T^{8} )^{2} \)
5 \( ( 1 + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 206 T^{2} + p^{4} T^{4} )^{2}( 1 + 13154 T^{4} + p^{8} T^{8} ) \)
13 \( ( 1 + p^{8} T^{8} )^{2} \)
17 \( ( 1 + p^{2} T^{2} )^{8} \)
19 \( ( 1 + p^{8} T^{8} )^{2} \)
23 \( ( 1 + 18 T + p^{2} T^{2} )^{4}( 1 - 734 T^{2} + p^{4} T^{4} )^{2} \)
29 \( ( 1 + 1234 T^{2} + p^{4} T^{4} )^{2}( 1 + 108194 T^{4} + p^{8} T^{8} ) \)
31 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
37 \( ( 1 - 1294 T^{2} + p^{4} T^{4} )^{2}( 1 - 2073886 T^{4} + p^{8} T^{8} ) \)
41 \( ( 1 + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 58 T + p^{2} T^{2} )^{4}( 1 - 6726046 T^{4} + p^{8} T^{8} ) \)
47 \( ( 1 + p^{2} T^{2} )^{8} \)
53 \( ( 1 - 6 T + p^{2} T^{2} )^{4}( 1 + 15377762 T^{4} + p^{8} T^{8} ) \)
59 \( ( 1 + p^{8} T^{8} )^{2} \)
61 \( ( 1 + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 118 T + p^{2} T^{2} )^{4}( 1 - 15839326 T^{4} + p^{8} T^{8} ) \)
71 \( ( 1 - 42331966 T^{4} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 64606846 T^{4} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + p^{8} T^{8} )^{2} \)
89 \( ( 1 + p^{4} T^{4} )^{4} \)
97 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
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

−5.35459421681946259652703290606, −5.13848750672266783615886887979, −4.76365539647367091052042110830, −4.73764933412013642423289347243, −4.71807783612912724964773627436, −4.50614807743334454131192218755, −4.46925210215706803719103557867, −4.18264968019418156429204174567, −4.10451022398143114593431585095, −3.76044708268531624241938622400, −3.42722133576052370437153103123, −3.37365948929583932237905658919, −3.30547028396814748049010646527, −3.16097539221147259327103074883, −3.05884279588131679403929938856, −2.95829285055756063958438994847, −2.43980599056720149603566143360, −1.97881038624140451074088826742, −1.84367102421015085516688006269, −1.81881636968344228230828217856, −1.69139949105388682926758068511, −1.62362042559376491096866370713, −0.64674409885536234715189849728, −0.64497943976343179949334934172, −0.39628387631228295883303581626, 0.39628387631228295883303581626, 0.64497943976343179949334934172, 0.64674409885536234715189849728, 1.62362042559376491096866370713, 1.69139949105388682926758068511, 1.81881636968344228230828217856, 1.84367102421015085516688006269, 1.97881038624140451074088826742, 2.43980599056720149603566143360, 2.95829285055756063958438994847, 3.05884279588131679403929938856, 3.16097539221147259327103074883, 3.30547028396814748049010646527, 3.37365948929583932237905658919, 3.42722133576052370437153103123, 3.76044708268531624241938622400, 4.10451022398143114593431585095, 4.18264968019418156429204174567, 4.46925210215706803719103557867, 4.50614807743334454131192218755, 4.71807783612912724964773627436, 4.73764933412013642423289347243, 4.76365539647367091052042110830, 5.13848750672266783615886887979, 5.35459421681946259652703290606

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

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