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Properties

Label 16-1183e8-1.1-c0e8-0-0
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  − 4·3-s + 10·9-s − 16-s − 24·27-s + 4·48-s − 4·53-s − 4·61-s + 4·79-s + 51·81-s − 4·107-s + 127-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 16·159-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·183-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4·3-s + 10·9-s − 16-s − 24·27-s + 4·48-s − 4·53-s − 4·61-s + 4·79-s + 51·81-s − 4·107-s + 127-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 16·159-s + 163-s + 167-s + 173-s + 179-s + 181-s + 16·183-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

\(L(\frac{1}{2})\) \(\approx\) \(0.01368585878\)
\(L(\frac12)\) \(\approx\) \(0.01368585878\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.52374474167369936385697124555, −4.42292790004324743908494914586, −4.20377575947168919511508372531, −4.09185815997014302375133492562, −3.95161560093229464119157194336, −3.77720189790728606583070336300, −3.76761433296570823770375489454, −3.71964309371922638137633649484, −3.51691339054230483958858266194, −3.25472734284793241518151277164, −3.22550849743973195031369652481, −2.99235024046009768568421986929, −2.88285712957823136508452782717, −2.52585554381448336925430007356, −2.40607410544921922877263731024, −2.24510670294923050317986348610, −2.03311860116144526847391745059, −1.96519689591437481652515371523, −1.72537873155947003046391856339, −1.57031237878080652670915735767, −1.48680335262374602522868620365, −1.30366030796550385958362119253, −1.14069724463199393431990080893, −0.74798229524772337485337226078, −0.12023736662908658976329524380, 0.12023736662908658976329524380, 0.74798229524772337485337226078, 1.14069724463199393431990080893, 1.30366030796550385958362119253, 1.48680335262374602522868620365, 1.57031237878080652670915735767, 1.72537873155947003046391856339, 1.96519689591437481652515371523, 2.03311860116144526847391745059, 2.24510670294923050317986348610, 2.40607410544921922877263731024, 2.52585554381448336925430007356, 2.88285712957823136508452782717, 2.99235024046009768568421986929, 3.22550849743973195031369652481, 3.25472734284793241518151277164, 3.51691339054230483958858266194, 3.71964309371922638137633649484, 3.76761433296570823770375489454, 3.77720189790728606583070336300, 3.95161560093229464119157194336, 4.09185815997014302375133492562, 4.20377575947168919511508372531, 4.42292790004324743908494914586, 4.52374474167369936385697124555

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

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