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Properties

Label 16-2960e8-1.1-c0e8-0-0
Degree $16$
Conductor $5.893\times 10^{27}$
Sign $1$
Analytic cond. $22.6772$
Root an. cond. $1.21541$
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  + 8·11-s + 16-s + 4·19-s − 4·61-s + 8·71-s − 81-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 32·209-s + 211-s + 223-s + ⋯
L(s)  = 1  + 8·11-s + 16-s + 4·19-s − 4·61-s + 8·71-s − 81-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 8·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 32·209-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 37^{8}\)
Sign: $1$
Analytic conductor: \(22.6772\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 37^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.690999458\)
\(L(\frac12)\) \(\approx\) \(8.690999458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.64203260469956747632775179996, −3.62016131777503598570947721219, −3.59068023321267912841357674426, −3.43263728373807358868322436743, −3.42590095139242383781322223446, −3.37372089916891360089485914056, −3.36587358769021412661566071832, −3.35795591282113823931740057784, −3.19170597788493676362576017763, −2.66715864864901269593303929649, −2.65800203390311993806934389188, −2.39861263234242801513711742782, −2.35229586414907195073418928406, −2.32077407401874599878841050765, −2.18044004169467190790894783262, −1.87080098186009620701592786835, −1.64905434566287743034534347852, −1.55555278687904423799558057748, −1.51991581921127899845503661196, −1.34080451865814446633015855327, −1.15087998478333885996436097215, −1.06507529529319015343154828045, −1.04632468294480126495961989840, −0.922077117190105643590034155688, −0.833364628418546272467392375680, 0.833364628418546272467392375680, 0.922077117190105643590034155688, 1.04632468294480126495961989840, 1.06507529529319015343154828045, 1.15087998478333885996436097215, 1.34080451865814446633015855327, 1.51991581921127899845503661196, 1.55555278687904423799558057748, 1.64905434566287743034534347852, 1.87080098186009620701592786835, 2.18044004169467190790894783262, 2.32077407401874599878841050765, 2.35229586414907195073418928406, 2.39861263234242801513711742782, 2.65800203390311993806934389188, 2.66715864864901269593303929649, 3.19170597788493676362576017763, 3.35795591282113823931740057784, 3.36587358769021412661566071832, 3.37372089916891360089485914056, 3.42590095139242383781322223446, 3.43263728373807358868322436743, 3.59068023321267912841357674426, 3.62016131777503598570947721219, 3.64203260469956747632775179996

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

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