Properties

Label 16-2816e8-1.1-c1e8-0-2
Degree $16$
Conductor $3.954\times 10^{27}$
Sign $1$
Analytic cond. $6.53548\times 10^{10}$
Root an. cond. $4.74192$
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  − 12·9-s + 36·25-s − 24·49-s + 54·81-s + 8·89-s + 56·97-s − 40·113-s − 20·121-s + ⋯
L(s)  = 1  − 4·9-s + 36/5·25-s − 3.42·49-s + 6·81-s + 0.847·89-s + 5.68·97-s − 3.76·113-s − 1.81·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(6.53548\times 10^{10}\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8587966647\)
\(L(\frac12)\) \(\approx\) \(0.8587966647\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
good3 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
5 \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \)
7 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + p T^{2} )^{8} \)
31 \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \)
37 \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 115 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - p T^{2} )^{8} \)
89 \( ( 1 - T + p T^{2} )^{8} \)
97 \( ( 1 - 7 T + p T^{2} )^{8} \)
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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

−3.56768773233725867129128592587, −3.48492457192287761857846995282, −3.32689703760533781902228241288, −3.27947515939082074040657678316, −3.20843796761174913293009131326, −3.07508038909512061510742577332, −2.79761226668416507967342909216, −2.77102926840342721392105049490, −2.69680237253423014386167440856, −2.66713021319210288931251446313, −2.62053335104647714710220497458, −2.35644119655056678295309102120, −2.32142115787033194405255948296, −2.15673895153154510300044245108, −2.01708819708109692954564936469, −1.76680142225483998983371338343, −1.37764912964072327380079048015, −1.36464885645486624498032389827, −1.22166837820830406477288991331, −1.14018886689054255854669735560, −1.02785839581326848966247950925, −0.923929144017714830669273293190, −0.39716105346649715937193441987, −0.29623884404025229383282503047, −0.13640364453173341195965038098, 0.13640364453173341195965038098, 0.29623884404025229383282503047, 0.39716105346649715937193441987, 0.923929144017714830669273293190, 1.02785839581326848966247950925, 1.14018886689054255854669735560, 1.22166837820830406477288991331, 1.36464885645486624498032389827, 1.37764912964072327380079048015, 1.76680142225483998983371338343, 2.01708819708109692954564936469, 2.15673895153154510300044245108, 2.32142115787033194405255948296, 2.35644119655056678295309102120, 2.62053335104647714710220497458, 2.66713021319210288931251446313, 2.69680237253423014386167440856, 2.77102926840342721392105049490, 2.79761226668416507967342909216, 3.07508038909512061510742577332, 3.20843796761174913293009131326, 3.27947515939082074040657678316, 3.32689703760533781902228241288, 3.48492457192287761857846995282, 3.56768773233725867129128592587

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

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