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Properties

Label 16-1575e8-1.1-c0e8-0-1
Degree $16$
Conductor $3.787\times 10^{25}$
Sign $1$
Analytic cond. $0.145714$
Root an. cond. $0.886581$
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  − 12·11-s + 2·16-s − 2·81-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 12·11-s + 2·16-s − 2·81-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.145714\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04841653473\)
\(L(\frac12)\) \(\approx\) \(0.04841653473\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
7 \( 1 - T^{4} + T^{8} \)
good2 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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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

−4.35882998953907408833590250625, −4.09496224878776000375739147809, −3.91039758810007255513451914938, −3.85822888133330841843386046942, −3.78803479193688569225931744245, −3.31253900829845018435878211664, −3.29643622920929884345041755892, −3.07908653980604714570524055063, −3.07280553664475052559952547976, −2.97688283374728085744946705221, −2.93477532665304763431442497267, −2.90261994390072871772686459368, −2.88155032220763718206105915475, −2.49056154912144387516855661431, −2.32056093655792188149513431544, −2.28340375124687014419468278309, −2.18651234074069398705553299977, −2.17832147218991758898316387840, −2.04931120395844944293695691713, −1.75980014867132028017009600836, −1.46692694034805235422491292652, −1.18192941330395818972174833013, −0.846054147690045600057702797730, −0.57229582545154634218202707623, −0.15858443668964376929472335757, 0.15858443668964376929472335757, 0.57229582545154634218202707623, 0.846054147690045600057702797730, 1.18192941330395818972174833013, 1.46692694034805235422491292652, 1.75980014867132028017009600836, 2.04931120395844944293695691713, 2.17832147218991758898316387840, 2.18651234074069398705553299977, 2.28340375124687014419468278309, 2.32056093655792188149513431544, 2.49056154912144387516855661431, 2.88155032220763718206105915475, 2.90261994390072871772686459368, 2.93477532665304763431442497267, 2.97688283374728085744946705221, 3.07280553664475052559952547976, 3.07908653980604714570524055063, 3.29643622920929884345041755892, 3.31253900829845018435878211664, 3.78803479193688569225931744245, 3.85822888133330841843386046942, 3.91039758810007255513451914938, 4.09496224878776000375739147809, 4.35882998953907408833590250625

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

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