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Properties

Label 16-525e8-1.1-c0e8-0-0
Degree $16$
Conductor $5.771\times 10^{21}$
Sign $1$
Analytic cond. $2.22091\times 10^{-5}$
Root an. cond. $0.511868$
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  + 2·16-s − 12·61-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + ⋯
L(s)  = 1  + 2·16-s − 12·61-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.22091\times 10^{-5}\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3406615068\)
\(L(\frac12)\) \(\approx\) \(0.3406615068\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−5.14959005626572939246939861449, −4.71787999659229431119451120672, −4.71368681971134051877329674142, −4.50364774584328870421161627358, −4.50189451263717730946963683119, −4.42602831652703613530353595059, −4.37837114333564791038553300389, −4.07759666830518613715473251466, −3.91490788644184588606427867294, −3.72425329598832982644511654373, −3.34047050755471509032231667467, −3.31966094801932222018327337308, −3.28412892766438771131336611196, −3.28069687518321590505497826757, −3.16921841480485411844525611973, −2.69936208651861485005943151331, −2.68599220391535101980008274751, −2.58214667023386001339226008871, −2.36390133880571560857729700542, −1.93422821101657829483960274435, −1.75017213250817479610146388580, −1.55080816345337616069209461415, −1.48726815656126762725697522187, −1.34065392843507600280010637163, −0.957285847337279441362749789493, 0.957285847337279441362749789493, 1.34065392843507600280010637163, 1.48726815656126762725697522187, 1.55080816345337616069209461415, 1.75017213250817479610146388580, 1.93422821101657829483960274435, 2.36390133880571560857729700542, 2.58214667023386001339226008871, 2.68599220391535101980008274751, 2.69936208651861485005943151331, 3.16921841480485411844525611973, 3.28069687518321590505497826757, 3.28412892766438771131336611196, 3.31966094801932222018327337308, 3.34047050755471509032231667467, 3.72425329598832982644511654373, 3.91490788644184588606427867294, 4.07759666830518613715473251466, 4.37837114333564791038553300389, 4.42602831652703613530353595059, 4.50189451263717730946963683119, 4.50364774584328870421161627358, 4.71368681971134051877329674142, 4.71787999659229431119451120672, 5.14959005626572939246939861449

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

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