Properties

Label 16-1045e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.422\times 10^{24}$
Sign $1$
Analytic cond. $0.00547251$
Root an. cond. $0.722165$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 8·7-s + 16-s − 2·17-s − 2·19-s + 2·23-s + 25-s + 16·35-s + 2·43-s − 2·47-s + 37·49-s + 2·73-s − 2·80-s + 81-s − 2·83-s + 4·85-s + 4·95-s − 8·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 2·5-s − 8·7-s + 16-s − 2·17-s − 2·19-s + 2·23-s + 25-s + 16·35-s + 2·43-s − 2·47-s + 37·49-s + 2·73-s − 2·80-s + 81-s − 2·83-s + 4·85-s + 4·95-s − 8·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.00547251\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1045} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004065253934\)
\(L(\frac12)\) \(\approx\) \(0.004065253934\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
17 \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
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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

−4.30086756444118193526864531424, −4.18044667961218228684078735921, −4.08983570414524206480266263171, −4.02053273824161051808262000806, −3.91821926641694518179665960016, −3.88961080248027901762942480075, −3.58272328720687667997332056707, −3.52426926814187165538719840398, −3.37392278937676019395858358822, −3.34744496540404984192603111420, −3.20525080269204547743481928837, −3.17257571266069514786330980794, −3.08204040309303206533053353630, −2.75210331790125772987230562119, −2.74682795875298409837362136743, −2.36589506210677997749271771848, −2.31498609638475493110199832038, −2.27400980854831912583511349420, −2.21840498176827732095871074766, −2.20167155601630569231964382076, −1.26592378844397844517887311359, −1.09912894912133402956015690423, −1.08352969010111989399973448756, −0.59207095733602129139067829226, −0.083150517460411159550230369345, 0.083150517460411159550230369345, 0.59207095733602129139067829226, 1.08352969010111989399973448756, 1.09912894912133402956015690423, 1.26592378844397844517887311359, 2.20167155601630569231964382076, 2.21840498176827732095871074766, 2.27400980854831912583511349420, 2.31498609638475493110199832038, 2.36589506210677997749271771848, 2.74682795875298409837362136743, 2.75210331790125772987230562119, 3.08204040309303206533053353630, 3.17257571266069514786330980794, 3.20525080269204547743481928837, 3.34744496540404984192603111420, 3.37392278937676019395858358822, 3.52426926814187165538719840398, 3.58272328720687667997332056707, 3.88961080248027901762942480075, 3.91821926641694518179665960016, 4.02053273824161051808262000806, 4.08983570414524206480266263171, 4.18044667961218228684078735921, 4.30086756444118193526864531424

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

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