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Properties

Label 16-3520e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.357\times 10^{28}$
Sign $1$
Analytic cond. $90.6981$
Root an. cond. $1.32540$
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  − 2·5-s + 2·9-s − 10·13-s + 25-s − 4·37-s − 10·41-s − 4·45-s − 3·49-s − 4·53-s + 20·65-s + 81-s + 4·89-s − 20·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 53·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·5-s + 2·9-s − 10·13-s + 25-s − 4·37-s − 10·41-s − 4·45-s − 3·49-s − 4·53-s + 20·65-s + 81-s + 4·89-s − 20·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 53·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(90.6981\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0006909476771\)
\(L(\frac12)\) \(\approx\) \(0.0006909476771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
7 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
13 \( ( 1 + T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
41 \( ( 1 + T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−3.64740821405852015648771336528, −3.63703840961434413446293865198, −3.61594853757396052553547754900, −3.33132410371803795576221888853, −3.27309762762276947669885530253, −3.20559759854973202297456463790, −2.98984149536575396436818030546, −2.97747101161077928289439014933, −2.91882231587356650098016040788, −2.77824653911763807307119551969, −2.58537468832784104712722878660, −2.48780750996050953198178367101, −2.36264841720141958517313904397, −2.08191842264885314374128726280, −1.89741411299580832486047724925, −1.81843844307068681782636688213, −1.77257610913343974952496459601, −1.76636677620376176427605368020, −1.72166063525341123343706409228, −1.61942926615745129772109401893, −1.51638881813148170907496236433, −0.74149996240054139839901668827, −0.65514776815750838012169449112, −0.12453172251733138432658917207, −0.04486708043886707124815792540, 0.04486708043886707124815792540, 0.12453172251733138432658917207, 0.65514776815750838012169449112, 0.74149996240054139839901668827, 1.51638881813148170907496236433, 1.61942926615745129772109401893, 1.72166063525341123343706409228, 1.76636677620376176427605368020, 1.77257610913343974952496459601, 1.81843844307068681782636688213, 1.89741411299580832486047724925, 2.08191842264885314374128726280, 2.36264841720141958517313904397, 2.48780750996050953198178367101, 2.58537468832784104712722878660, 2.77824653911763807307119551969, 2.91882231587356650098016040788, 2.97747101161077928289439014933, 2.98984149536575396436818030546, 3.20559759854973202297456463790, 3.27309762762276947669885530253, 3.33132410371803795576221888853, 3.61594853757396052553547754900, 3.63703840961434413446293865198, 3.64740821405852015648771336528

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

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