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  − 2·3-s + 4·7-s + 3·9-s − 8·21-s − 2·23-s − 2·27-s + 6·29-s − 4·43-s − 4·47-s + 6·49-s − 2·61-s + 12·63-s + 4·69-s + 81-s + 2·83-s − 12·87-s − 8·101-s + 6·103-s − 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s − 12·147-s + ⋯
L(s)  = 1  − 2·3-s + 4·7-s + 3·9-s − 8·21-s − 2·23-s − 2·27-s + 6·29-s − 4·43-s − 4·47-s + 6·49-s − 2·61-s + 12·63-s + 4·69-s + 81-s + 2·83-s − 12·87-s − 8·101-s + 6·103-s − 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s − 12·147-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  =  $(16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $1.480445608e-5$
$L(\frac12)$  $\approx$  $1.480445608e-5$
$L(1)$   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 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
23 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
29 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
37 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
53 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−3.72543614723940592243025582855, −3.70951708029968022373038195566, −3.40705796161011591993973174261, −3.36669611713169556130848060550, −3.31378896328819707873125047547, −3.12934380897232612678322026610, −2.93147685622055406538461647097, −2.81318755090589922217851250374, −2.71172869604517473426881947558, −2.60929555515067137449739993231, −2.56797105004540820397226871596, −2.43685760503564186696678873216, −2.24978459082322600216950942499, −1.97322287210696176740920862652, −1.80453810769698457888590891357, −1.78685198367493889744002326325, −1.77112715521194536998217613180, −1.49102229140475221554055949740, −1.32194223689711514608347402316, −1.29783510108590529105028596367, −1.21359031867850307218632711946, −1.21353372159341419248320091114, −1.04167940902770683982657533805, −0.67883757351304709131047255957, −0.000846644362155833575642995893, 0.000846644362155833575642995893, 0.67883757351304709131047255957, 1.04167940902770683982657533805, 1.21353372159341419248320091114, 1.21359031867850307218632711946, 1.29783510108590529105028596367, 1.32194223689711514608347402316, 1.49102229140475221554055949740, 1.77112715521194536998217613180, 1.78685198367493889744002326325, 1.80453810769698457888590891357, 1.97322287210696176740920862652, 2.24978459082322600216950942499, 2.43685760503564186696678873216, 2.56797105004540820397226871596, 2.60929555515067137449739993231, 2.71172869604517473426881947558, 2.81318755090589922217851250374, 2.93147685622055406538461647097, 3.12934380897232612678322026610, 3.31378896328819707873125047547, 3.36669611713169556130848060550, 3.40705796161011591993973174261, 3.70951708029968022373038195566, 3.72543614723940592243025582855

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

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