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Properties

Label 16-2700e8-1.1-c0e8-0-3
Degree $16$
Conductor $2.824\times 10^{27}$
Sign $1$
Analytic cond. $10.8684$
Root an. cond. $1.16080$
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  + 16-s + 12·41-s − 4·61-s + 4·121-s + 127-s + 131-s + 137-s + 139-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 + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 16-s + 12·41-s − 4·61-s + 4·121-s + 127-s + 131-s + 137-s + 139-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 + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{24} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(10.8684\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{24} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.632543131\)
\(L(\frac12)\) \(\approx\) \(2.632543131\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
5 \( 1 \)
good7 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
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

−3.93865983345903767771622835380, −3.76649022779146247843532120830, −3.73576919511552377909118070520, −3.63768598044373356370222390827, −3.47922817753105643036856906429, −3.24496340738967928274563252707, −3.02103935700100006035127423170, −3.00978871458902386970819184807, −2.93920836348384364619851783390, −2.90966193521288058435194783507, −2.64844653282263716906193318906, −2.63007653405636122465019883807, −2.39307970492415788101627758867, −2.25737931383648473236636789707, −2.25078462110706572923472945314, −2.12808453475780016851709470109, −2.03152601770716731794644619584, −1.77470614849101679548409762140, −1.47394562318368892549207236174, −1.23617580004521859076255459043, −1.21944424120963385988471203661, −1.10694678631106656372051560282, −0.998476391890196858342082375831, −0.72258602158063643159381688041, −0.55084722913320722797826727423, 0.55084722913320722797826727423, 0.72258602158063643159381688041, 0.998476391890196858342082375831, 1.10694678631106656372051560282, 1.21944424120963385988471203661, 1.23617580004521859076255459043, 1.47394562318368892549207236174, 1.77470614849101679548409762140, 2.03152601770716731794644619584, 2.12808453475780016851709470109, 2.25078462110706572923472945314, 2.25737931383648473236636789707, 2.39307970492415788101627758867, 2.63007653405636122465019883807, 2.64844653282263716906193318906, 2.90966193521288058435194783507, 2.93920836348384364619851783390, 3.00978871458902386970819184807, 3.02103935700100006035127423170, 3.24496340738967928274563252707, 3.47922817753105643036856906429, 3.63768598044373356370222390827, 3.73576919511552377909118070520, 3.76649022779146247843532120830, 3.93865983345903767771622835380

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

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