Properties

Label 16-552e8-1.1-c1e8-0-0
Degree $16$
Conductor $8.620\times 10^{21}$
Sign $1$
Analytic cond. $142471.$
Root an. cond. $2.09946$
Motivic weight $1$
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  − 4·3-s + 8·9-s − 8·13-s − 28·25-s − 20·27-s − 48·31-s + 32·39-s + 48·49-s − 40·73-s + 112·75-s + 50·81-s + 192·93-s − 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s − 192·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s − 2.21·13-s − 5.59·25-s − 3.84·27-s − 8.62·31-s + 5.12·39-s + 48/7·49-s − 4.68·73-s + 12.9·75-s + 50/9·81-s + 19.9·93-s − 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.8·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(142471.\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2486687741\)
\(L(\frac12)\) \(\approx\) \(0.2486687741\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23 \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
good5 \( ( 1 + 14 T^{2} + 94 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
7 \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + 14 T^{2} + 166 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 16 T^{2} + 322 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 6 T^{2} - 394 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 88 T^{2} + 3438 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + p T^{2} )^{8} \)
37 \( ( 1 + 26 T^{2} + 2062 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 92 T^{2} + 4198 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 38 T^{2} + 2934 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 206 T^{2} + 16222 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 174 T^{2} + 13886 T^{4} - 174 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 214 T^{2} + 20182 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 32 T^{2} + 9838 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 120 T^{2} + 13202 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 246 T^{2} + 28502 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 320 T^{2} + 41122 T^{4} + 320 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 108 T^{2} + 3734 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.68119015946928778833445014550, −4.52927282664644909688462798321, −4.47818695718296167966229949562, −4.43329276268403949784208836061, −4.10580493295389063352061380055, −3.98369827803966235198981914841, −3.96538771151568190099989655306, −3.74188980756035717074216153232, −3.66554235226217017543402643431, −3.57881652777935217480998624550, −3.48562584613584815460545328866, −3.23988662914998256573579790462, −3.03413336912760846863473329961, −2.47288230145750510600577473624, −2.45928763435267401814922324179, −2.43448093473988784236797999493, −2.15564736879325964523126007044, −2.05940171217713392051953755433, −1.76448485705016061065564578608, −1.67504871494726200973639065443, −1.66714262347443366177864946724, −1.35672409607243476044484771335, −0.43322026802330067264163923141, −0.35858432026172426610123530756, −0.34409592766589972110314568277, 0.34409592766589972110314568277, 0.35858432026172426610123530756, 0.43322026802330067264163923141, 1.35672409607243476044484771335, 1.66714262347443366177864946724, 1.67504871494726200973639065443, 1.76448485705016061065564578608, 2.05940171217713392051953755433, 2.15564736879325964523126007044, 2.43448093473988784236797999493, 2.45928763435267401814922324179, 2.47288230145750510600577473624, 3.03413336912760846863473329961, 3.23988662914998256573579790462, 3.48562584613584815460545328866, 3.57881652777935217480998624550, 3.66554235226217017543402643431, 3.74188980756035717074216153232, 3.96538771151568190099989655306, 3.98369827803966235198981914841, 4.10580493295389063352061380055, 4.43329276268403949784208836061, 4.47818695718296167966229949562, 4.52927282664644909688462798321, 4.68119015946928778833445014550

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

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