Properties

Degree 16
Conductor $ 2^{40} \cdot 5^{24} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·13-s + 2·17-s + 8·37-s + 8·53-s − 2·73-s + 81-s + 10·89-s − 2·97-s + 4·101-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·13-s + 2·17-s + 8·37-s + 8·53-s − 2·73-s + 81-s + 10·89-s − 2·97-s + 4·101-s + 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 5^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{4000} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $7.297684749$
$L(\frac12)$  $\approx$  $7.297684749$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
53 \( ( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.78078493873032308245740174740, −3.52244590849793936171477582073, −3.48444567854195544248087335568, −3.31183418884635302665518492425, −3.16506333841467406646810723115, −3.15585883393548520857470050963, −3.08297345838531656918322175391, −2.99578827622824067597478935302, −2.90822105014798316506708198786, −2.48985672056177143594054329864, −2.37278246452391450312378259737, −2.25216586337053787055338854721, −2.21647296787341595526501109389, −2.19462611321061363461237086159, −2.18089425945419952803428963485, −2.14485206147288123826099364059, −2.02921698656194227111275542751, −1.45272634367953630068183288294, −1.17462569582701276217779496759, −1.11519974014613452199786838663, −1.09943522073885087795436848309, −1.06279979876293503187293731204, −0.876628085897858628262699573574, −0.796156898929570856522469151921, −0.69952285132200704004475906738, 0.69952285132200704004475906738, 0.796156898929570856522469151921, 0.876628085897858628262699573574, 1.06279979876293503187293731204, 1.09943522073885087795436848309, 1.11519974014613452199786838663, 1.17462569582701276217779496759, 1.45272634367953630068183288294, 2.02921698656194227111275542751, 2.14485206147288123826099364059, 2.18089425945419952803428963485, 2.19462611321061363461237086159, 2.21647296787341595526501109389, 2.25216586337053787055338854721, 2.37278246452391450312378259737, 2.48985672056177143594054329864, 2.90822105014798316506708198786, 2.99578827622824067597478935302, 3.08297345838531656918322175391, 3.15585883393548520857470050963, 3.16506333841467406646810723115, 3.31183418884635302665518492425, 3.48444567854195544248087335568, 3.52244590849793936171477582073, 3.78078493873032308245740174740

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

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