Properties

Label 16-1368e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.227\times 10^{25}$
Sign $1$
Analytic cond. $0.0472004$
Root an. cond. $0.826269$
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  − 8·7-s + 16-s + 4·25-s + 8·31-s + 28·49-s + 4·73-s − 4·79-s − 8·97-s − 8·103-s − 8·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 8·7-s + 16-s + 4·25-s + 8·31-s + 28·49-s + 4·73-s − 4·79-s − 8·97-s − 8·103-s − 8·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.0472004\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1368} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03197544857\)
\(L(\frac12)\) \(\approx\) \(0.03197544857\)
\(L(1)\) 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 \( 1 \)
19 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 - T^{2} + T^{4} )^{4} \)
7 \( ( 1 + T + T^{2} )^{8} \)
11 \( ( 1 + T^{4} )^{4} \)
13 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 - T + T^{2} )^{8} \)
37 \( ( 1 - T^{2} + T^{4} )^{4} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} )^{4} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
71 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
73 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
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

−4.27142567997613092875469612002, −4.02882418011929879375592175176, −3.99360414659820788064266153440, −3.97806517890741944639866573825, −3.78394146839429030441905537050, −3.44161558601796967538204356814, −3.43896730823926328716090196817, −3.42204358598977679980193540041, −3.09047952805879842062926080407, −3.08273986455522512506891148363, −3.06827917098239451321384913778, −2.83745098967168823708320291643, −2.79484770197549617263948078139, −2.75534878326116380895287480850, −2.72317405135435340491006686293, −2.57061950849077855512158796412, −2.53257321682193062792767812108, −2.24081155348599780146611428612, −1.81321724579706628788823073695, −1.45157312927322497613592631426, −1.13076255205986667565091005338, −1.11498877643200776361233230571, −1.09987514374400221716913900359, −0.898426640689715484740847367470, −0.12219064685438725820572970355, 0.12219064685438725820572970355, 0.898426640689715484740847367470, 1.09987514374400221716913900359, 1.11498877643200776361233230571, 1.13076255205986667565091005338, 1.45157312927322497613592631426, 1.81321724579706628788823073695, 2.24081155348599780146611428612, 2.53257321682193062792767812108, 2.57061950849077855512158796412, 2.72317405135435340491006686293, 2.75534878326116380895287480850, 2.79484770197549617263948078139, 2.83745098967168823708320291643, 3.06827917098239451321384913778, 3.08273986455522512506891148363, 3.09047952805879842062926080407, 3.42204358598977679980193540041, 3.43896730823926328716090196817, 3.44161558601796967538204356814, 3.78394146839429030441905537050, 3.97806517890741944639866573825, 3.99360414659820788064266153440, 4.02882418011929879375592175176, 4.27142567997613092875469612002

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

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