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Properties

Label 16-2000e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.560\times 10^{26}$
Sign $1$
Analytic cond. $0.985137$
Root an. cond. $0.999064$
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  − 2·9-s + 4·29-s + 4·41-s + 8·49-s + 4·61-s + 81-s − 6·89-s − 4·101-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s + 4·29-s + 4·41-s + 8·49-s + 4·61-s + 81-s − 6·89-s − 4·101-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(0.985137\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330785700\)
\(L(\frac12)\) \(\approx\) \(1.330785700\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 - T )^{8}( 1 + T )^{8} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−4.19422591319230789415155372944, −3.95518872968926413251546852312, −3.84127804581096523936710819011, −3.74935178407869263455676998620, −3.71454461794880612354555124373, −3.34210958477396673885473725894, −3.32792880088743525522245812877, −3.31713973014315745940680911331, −3.00869180905099925723995394103, −2.66956096770201140043368430675, −2.64436361687078856065697842366, −2.63253375070817346907288119099, −2.53539253877544544433823084921, −2.46649607334182755180961385837, −2.44942633265448784985370316401, −2.43648941956941320723457294503, −2.37992379491733733596521599836, −1.81641116715361197686119114472, −1.66763592369434827954065441367, −1.24532942899974192960484432409, −1.20726570994757818288573132535, −1.18301575992665757179305782362, −1.07947189691273046784297102945, −0.808859571408472468738623792432, −0.49435598869371392984368206503, 0.49435598869371392984368206503, 0.808859571408472468738623792432, 1.07947189691273046784297102945, 1.18301575992665757179305782362, 1.20726570994757818288573132535, 1.24532942899974192960484432409, 1.66763592369434827954065441367, 1.81641116715361197686119114472, 2.37992379491733733596521599836, 2.43648941956941320723457294503, 2.44942633265448784985370316401, 2.46649607334182755180961385837, 2.53539253877544544433823084921, 2.63253375070817346907288119099, 2.64436361687078856065697842366, 2.66956096770201140043368430675, 3.00869180905099925723995394103, 3.31713973014315745940680911331, 3.32792880088743525522245812877, 3.34210958477396673885473725894, 3.71454461794880612354555124373, 3.74935178407869263455676998620, 3.84127804581096523936710819011, 3.95518872968926413251546852312, 4.19422591319230789415155372944

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

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