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Properties

Label 16-1183e8-1.1-c0e8-0-2
Degree $16$
Conductor $3.836\times 10^{24}$
Sign $1$
Analytic cond. $0.0147616$
Root an. cond. $0.768370$
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  + 8·3-s + 28·9-s − 16-s + 48·27-s − 8·48-s − 4·53-s + 8·61-s + 4·79-s + 6·81-s − 4·107-s + 127-s + 131-s + 137-s + 139-s − 28·144-s + 149-s + 151-s + 157-s − 32·159-s + 163-s + 167-s + 173-s + 179-s + 181-s + 64·183-s + 191-s + 193-s + ⋯
L(s)  = 1  + 8·3-s + 28·9-s − 16-s + 48·27-s − 8·48-s − 4·53-s + 8·61-s + 4·79-s + 6·81-s − 4·107-s + 127-s + 131-s + 137-s + 139-s − 28·144-s + 149-s + 151-s + 157-s − 32·159-s + 163-s + 167-s + 173-s + 179-s + 181-s + 64·183-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{8} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(0.0147616\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{8} \cdot 13^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(9.226552049\)
\(L(\frac12)\) \(\approx\) \(9.226552049\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T^{4} + T^{8} \)
13 \( 1 \)
good2 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
3 \( ( 1 - T + T^{2} )^{8} \)
5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 - T^{4} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
59 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
61 \( ( 1 - T + T^{2} )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T^{4} + T^{8} )^{2} \)
73 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
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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

−4.11667764740247445051332252352, −4.00746021266438724729793804716, −3.99199745682297909546108246992, −3.86395391679438149350426209554, −3.85685273428276413486683985111, −3.66618809010245038556753066462, −3.37312996470336227956669451231, −3.28907072261221640157585805252, −3.28496397545506401662201132979, −3.27831131597596203387959979300, −3.01178440414669880513342892789, −2.92911980328115114204382566273, −2.84285487084676817991284901099, −2.79940455946519419911530112165, −2.54703483499142868448390636643, −2.35459878878601995888217739185, −2.34243760690916444074598983591, −2.17650547640900464162521045918, −2.12264451276257881912407701599, −2.03924699872086419519639677515, −1.94276762270377635687395344062, −1.71547530436726376240172844413, −1.49231615283735433111874369824, −1.23652749384278976547603113930, −0.69298927756968005901700468680, 0.69298927756968005901700468680, 1.23652749384278976547603113930, 1.49231615283735433111874369824, 1.71547530436726376240172844413, 1.94276762270377635687395344062, 2.03924699872086419519639677515, 2.12264451276257881912407701599, 2.17650547640900464162521045918, 2.34243760690916444074598983591, 2.35459878878601995888217739185, 2.54703483499142868448390636643, 2.79940455946519419911530112165, 2.84285487084676817991284901099, 2.92911980328115114204382566273, 3.01178440414669880513342892789, 3.27831131597596203387959979300, 3.28496397545506401662201132979, 3.28907072261221640157585805252, 3.37312996470336227956669451231, 3.66618809010245038556753066462, 3.85685273428276413486683985111, 3.86395391679438149350426209554, 3.99199745682297909546108246992, 4.00746021266438724729793804716, 4.11667764740247445051332252352

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

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