Properties

Label 16-1620e8-1.1-c1e8-0-1
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $7.84037\times 10^{8}$
Root an. cond. $3.59663$
Motivic weight $1$
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  − 12·19-s + 3·25-s − 4·31-s + 4·49-s + 28·61-s − 48·79-s − 56·109-s − 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2.75·19-s + 3/5·25-s − 0.718·31-s + 4/7·49-s + 3.58·61-s − 5.40·79-s − 5.36·109-s − 3.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(7.84037\times 10^{8}\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.05682076014\)
\(L(\frac12)\) \(\approx\) \(0.05682076014\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 3 T^{2} + 4 p T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
good7 \( ( 1 - 2 T^{2} - 30 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - T^{2} + 48 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 15 T^{2} + 602 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 24 T^{2} + 686 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 59 T^{2} + 3318 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 129 T^{2} + 7232 T^{4} + 129 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 146 T^{2} + 8898 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 18 T^{2} + 1274 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 134 T^{2} + 8946 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 3 p T^{2} + 13988 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 7 T + 102 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 8 T^{2} - 3906 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 53 T^{2} + 3528 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 227 T^{2} + 22734 T^{4} - 227 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 6 T + p T^{2} )^{8} \)
83 \( ( 1 - 210 T^{2} + 23642 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 246 T^{2} + 28907 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 182 T^{2} + 16650 T^{4} - 182 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.97279057206456251125364689685, −3.92270048991999864574954141683, −3.80641421395681965773113392101, −3.74373155189888670241321788738, −3.61056285882823967062008888626, −3.14467889491210168575978120433, −3.10708088473412889446418952197, −3.07007998573965699000917187943, −2.98677255989246034600843760227, −2.80255466588719061263647788646, −2.74231423907335546240935448168, −2.48124376878164491183308734543, −2.31357399947324902391854163187, −2.27501439770638048899578206733, −2.08462541596955301521802529456, −2.04120237246042569486673448670, −1.97695313580484505490615316462, −1.52450434883807959672339501111, −1.31281490460219658780685591310, −1.29008833056833967569872976249, −1.21365738161449770588939694300, −1.05421333218261711688018379341, −0.69088774826727342543938868381, −0.14230163989165245450413356432, −0.06794938029132629590035500001, 0.06794938029132629590035500001, 0.14230163989165245450413356432, 0.69088774826727342543938868381, 1.05421333218261711688018379341, 1.21365738161449770588939694300, 1.29008833056833967569872976249, 1.31281490460219658780685591310, 1.52450434883807959672339501111, 1.97695313580484505490615316462, 2.04120237246042569486673448670, 2.08462541596955301521802529456, 2.27501439770638048899578206733, 2.31357399947324902391854163187, 2.48124376878164491183308734543, 2.74231423907335546240935448168, 2.80255466588719061263647788646, 2.98677255989246034600843760227, 3.07007998573965699000917187943, 3.10708088473412889446418952197, 3.14467889491210168575978120433, 3.61056285882823967062008888626, 3.74373155189888670241321788738, 3.80641421395681965773113392101, 3.92270048991999864574954141683, 3.97279057206456251125364689685

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

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