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Properties

Label 16-1175e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.633\times 10^{24}$
Sign $1$
Analytic cond. $0.0139816$
Root an. cond. $0.765768$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 16-s + 36-s − 49-s − 2·59-s − 4·61-s + 2·71-s + 4·79-s + 81-s + 4·89-s + 2·101-s + 8·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s − 9-s + 16-s + 36-s − 49-s − 2·59-s − 4·61-s + 2·71-s + 4·79-s + 81-s + 4·89-s + 2·101-s + 8·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(0.0139816\)
Root analytic conductor: \(0.765768\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5573115437\)
\(L(\frac12)\) \(\approx\) \(0.5573115437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( ( 1 + T^{2} )^{4} \)
good2 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
3 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
7 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
11 \( ( 1 - T )^{8}( 1 + T )^{8} \)
13 \( ( 1 + T^{2} )^{8} \)
17 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
19 \( ( 1 - T )^{8}( 1 + T )^{8} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
73 \( ( 1 + T^{2} )^{8} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
83 \( ( 1 - T^{2} + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.52496732286942246736032786755, −4.26868305226516899442133904207, −4.26370006274438125092474720323, −3.92124481340279724041052241985, −3.78154919673813538518816429505, −3.75392496447895608194580141521, −3.52658839941228764247314321820, −3.50940431886982839368359529020, −3.39606757631291805847258580020, −3.38637430882455860604812618963, −3.23879783383826434540177440160, −2.97857011916711530973115478925, −2.92571644589700109699345556520, −2.55528303036111857217945985189, −2.51765944306258292725493912237, −2.36076373300112229943515110206, −2.22823615799587138708362448891, −2.18989955715817448199806730015, −1.80558718622154976639712892754, −1.66021656389120445258364584229, −1.57660892888746229596667437219, −1.45002622795896057191427139435, −0.908055115876889883477446276673, −0.829438365804314939009805640779, −0.61927679181721411447011770179, 0.61927679181721411447011770179, 0.829438365804314939009805640779, 0.908055115876889883477446276673, 1.45002622795896057191427139435, 1.57660892888746229596667437219, 1.66021656389120445258364584229, 1.80558718622154976639712892754, 2.18989955715817448199806730015, 2.22823615799587138708362448891, 2.36076373300112229943515110206, 2.51765944306258292725493912237, 2.55528303036111857217945985189, 2.92571644589700109699345556520, 2.97857011916711530973115478925, 3.23879783383826434540177440160, 3.38637430882455860604812618963, 3.39606757631291805847258580020, 3.50940431886982839368359529020, 3.52658839941228764247314321820, 3.75392496447895608194580141521, 3.78154919673813538518816429505, 3.92124481340279724041052241985, 4.26370006274438125092474720323, 4.26868305226516899442133904207, 4.52496732286942246736032786755

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

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