Properties

Label 16-44e16-1.1-c0e8-0-1
Degree $16$
Conductor $1.974\times 10^{26}$
Sign $1$
Analytic cond. $0.759451$
Root an. cond. $0.982949$
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  − 2·5-s − 2·9-s + 3·25-s + 2·37-s + 4·45-s − 2·49-s + 2·53-s + 81-s − 8·89-s − 2·97-s − 2·113-s − 2·125-s + ⋯
L(s)  = 1  − 2·5-s − 2·9-s + 3·25-s + 2·37-s + 4·45-s − 2·49-s + 2·53-s + 81-s − 8·89-s − 2·97-s − 2·113-s − 2·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(0.759451\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4602684080\)
\(L(\frac12)\) \(\approx\) \(0.4602684080\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
5 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
17 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T )^{8} \)
29 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
41 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
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 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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 \( ( 1 + T + T^{2} )^{8} \)
97 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
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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

−4.07553816715637678952533281306, −3.90494878932236980290845239602, −3.84716941755525948204438644497, −3.81880236807447309165194358646, −3.74991410118336632397622267533, −3.32191482626995293262723097491, −3.27716243791473090027925024668, −3.19445570054845118679756247912, −3.03874592509396175104927650682, −3.01975304059109014472348564431, −2.92688753634423534681233191806, −2.84591209564986645145065473418, −2.63058191468196140513303469363, −2.53143957115242267156232408194, −2.30314006194892615743890123915, −2.26173430705573573218969403253, −2.20649542018197951667892867321, −1.61804378891334899376336735800, −1.60377075098295251981798623456, −1.57175242118220004916614115878, −1.50905923397523293527971811182, −1.05833888453604244014523811685, −0.942358877864749815717422780218, −0.54483001185000933285870754486, −0.43790370867925483300503658614, 0.43790370867925483300503658614, 0.54483001185000933285870754486, 0.942358877864749815717422780218, 1.05833888453604244014523811685, 1.50905923397523293527971811182, 1.57175242118220004916614115878, 1.60377075098295251981798623456, 1.61804378891334899376336735800, 2.20649542018197951667892867321, 2.26173430705573573218969403253, 2.30314006194892615743890123915, 2.53143957115242267156232408194, 2.63058191468196140513303469363, 2.84591209564986645145065473418, 2.92688753634423534681233191806, 3.01975304059109014472348564431, 3.03874592509396175104927650682, 3.19445570054845118679756247912, 3.27716243791473090027925024668, 3.32191482626995293262723097491, 3.74991410118336632397622267533, 3.81880236807447309165194358646, 3.84716941755525948204438644497, 3.90494878932236980290845239602, 4.07553816715637678952533281306

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

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