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Properties

Label 16-539e8-1.1-c0e8-0-1
Degree $16$
Conductor $7.124\times 10^{21}$
Sign $1$
Analytic cond. $2.74136\times 10^{-5}$
Root an. cond. $0.518648$
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  + 2·2-s + 3·4-s + 2·8-s + 9-s + 11-s + 16-s + 2·18-s + 2·22-s + 2·23-s + 25-s − 4·29-s − 2·32-s + 3·36-s − 3·37-s − 4·43-s + 3·44-s + 4·46-s + 2·50-s − 3·53-s − 8·58-s − 4·64-s + 2·67-s + 6·71-s + 2·72-s − 6·74-s − 3·79-s + 81-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 2·8-s + 9-s + 11-s + 16-s + 2·18-s + 2·22-s + 2·23-s + 25-s − 4·29-s − 2·32-s + 3·36-s − 3·37-s − 4·43-s + 3·44-s + 4·46-s + 2·50-s − 3·53-s − 8·58-s − 4·64-s + 2·67-s + 6·71-s + 2·72-s − 6·74-s − 3·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.74136\times 10^{-5}\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{16} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228119359\)
\(L(\frac12)\) \(\approx\) \(1.228119359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
37 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
53 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
67 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
71 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
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

−5.00637823702467795140908556027, −4.96121847149860175418137347947, −4.87121119046073565200332816518, −4.68891199600432021625541514371, −4.41330667832761534174787731366, −4.30260520800787961699545555861, −4.15329554333321109679413915113, −3.89644361129510181897526199837, −3.72005970911072326777262703267, −3.61333444021680828668920483969, −3.58815081625988808005063088001, −3.54467650920945571614482761402, −3.38300823153009680500295158990, −3.28545146148892397233808562247, −3.18219858845019443251621199287, −2.86839505731870621128647741785, −2.57380846926098713644529290350, −2.55627314726638100053573131761, −2.31533184727614739137065433763, −2.11378030800102174452325582218, −1.95093617073094613198620587226, −1.56099157858285865479979560829, −1.52887838484128004326729690525, −1.40650063856481072446570543503, −1.27703244638978597176428685725, 1.27703244638978597176428685725, 1.40650063856481072446570543503, 1.52887838484128004326729690525, 1.56099157858285865479979560829, 1.95093617073094613198620587226, 2.11378030800102174452325582218, 2.31533184727614739137065433763, 2.55627314726638100053573131761, 2.57380846926098713644529290350, 2.86839505731870621128647741785, 3.18219858845019443251621199287, 3.28545146148892397233808562247, 3.38300823153009680500295158990, 3.54467650920945571614482761402, 3.58815081625988808005063088001, 3.61333444021680828668920483969, 3.72005970911072326777262703267, 3.89644361129510181897526199837, 4.15329554333321109679413915113, 4.30260520800787961699545555861, 4.41330667832761534174787731366, 4.68891199600432021625541514371, 4.87121119046073565200332816518, 4.96121847149860175418137347947, 5.00637823702467795140908556027

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

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