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Properties

Label 16-980e8-1.1-c0e8-0-2
Degree $16$
Conductor $8.508\times 10^{23}$
Sign $1$
Analytic cond. $0.00327390$
Root an. cond. $0.699345$
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  + 2·4-s + 4·9-s + 16-s + 8·36-s − 2·64-s + 6·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 2·4-s + 4·9-s + 16-s + 8·36-s − 2·64-s + 6·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(0.00327390\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{8} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.960573557\)
\(L(\frac12)\) \(\approx\) \(1.960573557\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.48927676494909329218206359569, −4.45990399303347121412776320637, −4.15730358470410986281333019978, −4.11615482734074465580480162201, −4.08533714243369585529571767341, −4.03137258369416807580917582221, −3.74268858308021409640504315515, −3.62206610801279245905909710501, −3.47096316806423292961315868201, −3.35668466705718864119965099261, −3.31666910974646657766852397326, −3.01497576584863335499380343426, −2.73960846638942985615717002133, −2.72931873858675939607097594529, −2.46523788384924324392822915304, −2.43702674696240688646207824865, −2.38419108559785237991325984166, −2.20521203852142335097996614980, −1.73813263557611878761106804737, −1.71814407002169286595163026182, −1.66565511279663989273621241601, −1.56997585376123356412284055248, −1.13762375495705468201852172298, −1.12503455848332174329274823098, −1.09103090404951139878806406777, 1.09103090404951139878806406777, 1.12503455848332174329274823098, 1.13762375495705468201852172298, 1.56997585376123356412284055248, 1.66565511279663989273621241601, 1.71814407002169286595163026182, 1.73813263557611878761106804737, 2.20521203852142335097996614980, 2.38419108559785237991325984166, 2.43702674696240688646207824865, 2.46523788384924324392822915304, 2.72931873858675939607097594529, 2.73960846638942985615717002133, 3.01497576584863335499380343426, 3.31666910974646657766852397326, 3.35668466705718864119965099261, 3.47096316806423292961315868201, 3.62206610801279245905909710501, 3.74268858308021409640504315515, 4.03137258369416807580917582221, 4.08533714243369585529571767341, 4.11615482734074465580480162201, 4.15730358470410986281333019978, 4.45990399303347121412776320637, 4.48927676494909329218206359569

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

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