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Properties

Label 20-712e10-1.1-c0e10-0-1
Degree $20$
Conductor $3.348\times 10^{28}$
Sign $1$
Analytic cond. $3.20901\times 10^{-5}$
Root an. cond. $0.596099$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 2·6-s + 9-s + 9·11-s + 9·17-s − 18-s − 2·19-s − 9·22-s − 25-s − 18·33-s − 9·34-s + 2·38-s − 2·41-s − 2·43-s − 49-s + 50-s − 18·51-s + 4·57-s − 2·59-s + 18·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s − 2·83-s + 2·86-s + ⋯
L(s)  = 1  − 2-s − 2·3-s + 2·6-s + 9-s + 9·11-s + 9·17-s − 18-s − 2·19-s − 9·22-s − 25-s − 18·33-s − 9·34-s + 2·38-s − 2·41-s − 2·43-s − 49-s + 50-s − 18·51-s + 4·57-s − 2·59-s + 18·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s − 2·83-s + 2·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 89^{10}\)
Sign: $1$
Analytic conductor: \(3.20901\times 10^{-5}\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 89^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2343791428\)
\(L(\frac12)\) \(\approx\) \(0.2343791428\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
good3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
11 \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
17 \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
37 \( ( 1 - T )^{10}( 1 + T )^{10} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.99004821180041182528631374807, −3.89664009241904208880875700200, −3.87865399898078605144766319093, −3.75832107948327206604236656655, −3.66459011618508755048608459373, −3.65193831737785199977724193417, −3.52056049804484945678933927214, −3.51714542184789561285887230215, −3.29016378712050823822261355823, −3.09418354843629659759596006517, −3.05212854071229786676559595183, −3.00455720293586337967634438441, −2.89516130504653282196291442676, −2.69010799696148106615355746845, −2.27178452513696149305453461663, −2.21048062695438297806811925026, −1.80711825186729283861718714979, −1.67365923784166771188416425982, −1.54382699945098472470060635177, −1.53299712250101336412350124626, −1.35933979859112496892744866388, −1.25713073863435973788773239179, −1.17257149839545828214491774049, −1.15159487801728084134311645291, −1.06949721861341796834878220543, 1.06949721861341796834878220543, 1.15159487801728084134311645291, 1.17257149839545828214491774049, 1.25713073863435973788773239179, 1.35933979859112496892744866388, 1.53299712250101336412350124626, 1.54382699945098472470060635177, 1.67365923784166771188416425982, 1.80711825186729283861718714979, 2.21048062695438297806811925026, 2.27178452513696149305453461663, 2.69010799696148106615355746845, 2.89516130504653282196291442676, 3.00455720293586337967634438441, 3.05212854071229786676559595183, 3.09418354843629659759596006517, 3.29016378712050823822261355823, 3.51714542184789561285887230215, 3.52056049804484945678933927214, 3.65193831737785199977724193417, 3.66459011618508755048608459373, 3.75832107948327206604236656655, 3.87865399898078605144766319093, 3.89664009241904208880875700200, 3.99004821180041182528631374807

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

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