Properties

Degree 16
Conductor $ 3^{8} \cdot 211^{8} $
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 − 3·4-s − 2·7-s + 9-s − 6·12-s + 2·13-s + 6·16-s − 2·19-s − 4·21-s − 25-s + 6·28-s − 4·31-s − 3·36-s + 4·39-s − 8·43-s + 12·48-s + 3·49-s − 6·52-s − 4·57-s − 6·61-s − 2·63-s − 10·64-s − 8·73-s − 2·75-s + 6·76-s + 12·84-s − 4·91-s + ⋯
L(s)  = 1  + 2·3-s − 3·4-s − 2·7-s + 9-s − 6·12-s + 2·13-s + 6·16-s − 2·19-s − 4·21-s − 25-s + 6·28-s − 4·31-s − 3·36-s + 4·39-s − 8·43-s + 12·48-s + 3·49-s − 6·52-s − 4·57-s − 6·61-s − 2·63-s − 10·64-s − 8·73-s − 2·75-s + 6·76-s + 12·84-s − 4·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 211^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 211^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{8} \cdot 211^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{633} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 3^{8} \cdot 211^{8} ,\ ( \ : [0]^{8} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.02296546352$
$L(\frac12)$  $\approx$  $0.02296546352$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;211\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{3,\;211\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
211 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
5 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
7 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
23 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
29 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
43 \( ( 1 + T + T^{2} )^{8} \)
47 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 + T + T^{2} )^{8} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
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

−4.89769086620307088203601061944, −4.73132513355169434220410220191, −4.44885057391912521562496132714, −4.39724917669075648191804250101, −4.25708827139220254382524781558, −4.19844806842361299171218274205, −3.90665175574649274161024980520, −3.78506068125476433788374115224, −3.68982596198123419253566341173, −3.68582118115226648217656685882, −3.64388995303213578945049454162, −3.17123121487354813499534491146, −3.14362762540509556208084181619, −3.10575332720359491674074358631, −3.04199242463974218999989252205, −2.90008629134652037831146540343, −2.81098382548455277816684619370, −2.57607739656722820028388863709, −2.00782974788260612474060929786, −1.77678632430681360936367034764, −1.68981350742257403663191874460, −1.59174564648898276770946992627, −1.55753865669831610511655494103, −1.33545731319641007077210955580, −0.15209118831223742791347645294, 0.15209118831223742791347645294, 1.33545731319641007077210955580, 1.55753865669831610511655494103, 1.59174564648898276770946992627, 1.68981350742257403663191874460, 1.77678632430681360936367034764, 2.00782974788260612474060929786, 2.57607739656722820028388863709, 2.81098382548455277816684619370, 2.90008629134652037831146540343, 3.04199242463974218999989252205, 3.10575332720359491674074358631, 3.14362762540509556208084181619, 3.17123121487354813499534491146, 3.64388995303213578945049454162, 3.68582118115226648217656685882, 3.68982596198123419253566341173, 3.78506068125476433788374115224, 3.90665175574649274161024980520, 4.19844806842361299171218274205, 4.25708827139220254382524781558, 4.39724917669075648191804250101, 4.44885057391912521562496132714, 4.73132513355169434220410220191, 4.89769086620307088203601061944

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

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