Properties

Label 16-3360e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.624\times 10^{28}$
Sign $1$
Analytic cond. $62.5131$
Root an. cond. $1.29493$
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  + 2·25-s + 81-s + 12·89-s − 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 2·25-s + 81-s + 12·89-s − 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(62.5131\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.272356889\)
\(L(\frac12)\) \(\approx\) \(2.272356889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{4} + T^{8} \)
5 \( ( 1 - T^{2} + T^{4} )^{2} \)
7 \( 1 - T^{4} + T^{8} \)
good11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( ( 1 - T^{2} + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
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

−3.78423519032791927515790176156, −3.67468599679942541534562726457, −3.54365403883430324923092362188, −3.38871198146869784701233093348, −3.28087587167552501671153106500, −3.26781626672289329755711746485, −3.16775500966879478819873456069, −2.79948632957793143150540680119, −2.77333756730893736275259929571, −2.70685446278997619806693725951, −2.65517240350598773487149643906, −2.60234222677032192678699537807, −2.50329197888624356224929205153, −2.17643689812507960871961271041, −1.98227904507205684182959423852, −1.90862499772350490124523117414, −1.76384034667518348539089468111, −1.76319210846271758349500120481, −1.67643038866331166870515898474, −1.31982921631893166719888410208, −1.11800789827827589191487110764, −1.04057673472036721774141244882, −0.841387865048662697780064150988, −0.71573768746270268383465830780, −0.42420924918273687271557559479, 0.42420924918273687271557559479, 0.71573768746270268383465830780, 0.841387865048662697780064150988, 1.04057673472036721774141244882, 1.11800789827827589191487110764, 1.31982921631893166719888410208, 1.67643038866331166870515898474, 1.76319210846271758349500120481, 1.76384034667518348539089468111, 1.90862499772350490124523117414, 1.98227904507205684182959423852, 2.17643689812507960871961271041, 2.50329197888624356224929205153, 2.60234222677032192678699537807, 2.65517240350598773487149643906, 2.70685446278997619806693725951, 2.77333756730893736275259929571, 2.79948632957793143150540680119, 3.16775500966879478819873456069, 3.26781626672289329755711746485, 3.28087587167552501671153106500, 3.38871198146869784701233093348, 3.54365403883430324923092362188, 3.67468599679942541534562726457, 3.78423519032791927515790176156

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

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