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Properties

Label 16-1008e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.066\times 10^{24}$
Sign $1$
Analytic cond. $0.00410148$
Root an. cond. $0.709265$
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 + 4·19-s − 4·31-s − 4·37-s − 8·43-s + 2·49-s − 4·79-s + 8·97-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 + ⋯
L(s)  = 1  + 16-s + 4·19-s − 4·31-s − 4·37-s − 8·43-s + 2·49-s − 4·79-s + 8·97-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.00410148\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5379388651\)
\(L(\frac12)\) \(\approx\) \(0.5379388651\)
\(L(1)\) 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 \( 1 \)
7 \( ( 1 - T^{2} + T^{4} )^{2} \)
good5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} )^{4} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
37 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
61 \( ( 1 - T^{4} + T^{8} )^{2} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T^{4} + T^{8} )^{2} \)
97 \( ( 1 - T + T^{2} )^{8} \)
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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

−4.56601589509895399925881928518, −4.39162977064372153874124114183, −4.29152361719826470835145568307, −4.09244344360642072801064585100, −3.84307985498089923388736742924, −3.67777513871499827172391819859, −3.67211504881991137074508897291, −3.55893614950757876022507068500, −3.36479585519464963054673514120, −3.30803971102687942955559735781, −3.26167031981951380052890179966, −3.23687082751312733694225799043, −3.10713870736498415462450695059, −2.90737221835995836305519861157, −2.64159478866121775166201069601, −2.31095785793344059921823399316, −2.27132686108002357233558864137, −2.03254602209798490097493974566, −1.78564265226672449530260624586, −1.76166584827434724853059176492, −1.59764136003858419182628434197, −1.49853715802956434103445936430, −1.20687041158891062408619798711, −1.15263949110931539728575839279, −0.48373112927566224127392078059, 0.48373112927566224127392078059, 1.15263949110931539728575839279, 1.20687041158891062408619798711, 1.49853715802956434103445936430, 1.59764136003858419182628434197, 1.76166584827434724853059176492, 1.78564265226672449530260624586, 2.03254602209798490097493974566, 2.27132686108002357233558864137, 2.31095785793344059921823399316, 2.64159478866121775166201069601, 2.90737221835995836305519861157, 3.10713870736498415462450695059, 3.23687082751312733694225799043, 3.26167031981951380052890179966, 3.30803971102687942955559735781, 3.36479585519464963054673514120, 3.55893614950757876022507068500, 3.67211504881991137074508897291, 3.67777513871499827172391819859, 3.84307985498089923388736742924, 4.09244344360642072801064585100, 4.29152361719826470835145568307, 4.39162977064372153874124114183, 4.56601589509895399925881928518

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

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