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Properties

Label 16-1900e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.698\times 10^{26}$
Sign $1$
Analytic cond. $0.653560$
Root an. cond. $0.973767$
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  + 5-s + 2·7-s − 2·9-s − 3·11-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 2·35-s + 2·43-s − 2·45-s + 2·47-s + 3·49-s − 3·55-s + 2·61-s − 4·63-s + 2·73-s − 6·77-s + 81-s + 6·83-s − 3·85-s − 2·95-s + 6·99-s − 4·101-s + 6·115-s − 6·119-s + 3·121-s + ⋯
L(s)  = 1  + 5-s + 2·7-s − 2·9-s − 3·11-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 2·35-s + 2·43-s − 2·45-s + 2·47-s + 3·49-s − 3·55-s + 2·61-s − 4·63-s + 2·73-s − 6·77-s + 81-s + 6·83-s − 3·85-s − 2·95-s + 6·99-s − 4·101-s + 6·115-s − 6·119-s + 3·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.653560\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{16} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104760077\)
\(L(\frac12)\) \(\approx\) \(1.104760077\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
11 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
23 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−4.10342165060886491190830641064, −3.91315109061995638198355966103, −3.88214336155628197450283314049, −3.73309554542912994467638384498, −3.67283033647532118699740048407, −3.52322342757056530851500717599, −3.24634210267969985332637367493, −3.18233632467201119368798440317, −3.11594554935862544148515280692, −2.62866972466538092760993707399, −2.62120132765566967346987971891, −2.56019610223348062616543685430, −2.53543804531689072451829621519, −2.48194514041375049682327861837, −2.46892298113747544709107222301, −2.41685540173557012219518480774, −2.25974538650321310691659273504, −1.93194513664864831252879003522, −1.71228496827695187931748051540, −1.64496422860681170422904918114, −1.24876908150718269162853781313, −1.14833579578359337394675630989, −0.956568679633846329940462433716, −0.912382448147095896730278444607, −0.40962753579880184930645718525, 0.40962753579880184930645718525, 0.912382448147095896730278444607, 0.956568679633846329940462433716, 1.14833579578359337394675630989, 1.24876908150718269162853781313, 1.64496422860681170422904918114, 1.71228496827695187931748051540, 1.93194513664864831252879003522, 2.25974538650321310691659273504, 2.41685540173557012219518480774, 2.46892298113747544709107222301, 2.48194514041375049682327861837, 2.53543804531689072451829621519, 2.56019610223348062616543685430, 2.62120132765566967346987971891, 2.62866972466538092760993707399, 3.11594554935862544148515280692, 3.18233632467201119368798440317, 3.24634210267969985332637367493, 3.52322342757056530851500717599, 3.67283033647532118699740048407, 3.73309554542912994467638384498, 3.88214336155628197450283314049, 3.91315109061995638198355966103, 4.10342165060886491190830641064

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

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