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Properties

Label 20-1035e10-1.1-c0e10-0-1
Degree $20$
Conductor $1.411\times 10^{30}$
Sign $1$
Analytic cond. $0.00135199$
Root an. cond. $0.718701$
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  − 4-s + 5-s + 9·17-s − 20-s − 23-s + 2·31-s + 49-s + 2·53-s − 9·68-s + 2·83-s + 9·85-s + 92-s − 2·107-s + 2·113-s − 115-s − 121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4-s + 5-s + 9·17-s − 20-s − 23-s + 2·31-s + 49-s + 2·53-s − 9·68-s + 2·83-s + 9·85-s + 92-s − 2·107-s + 2·113-s − 115-s − 121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{20} \cdot 5^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(0.00135199\)
Root analytic conductor: \(0.718701\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{20} \cdot 5^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.180075132\)
\(L(\frac12)\) \(\approx\) \(1.180075132\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
37 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
43 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
59 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
71 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
97 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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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

−3.74791741496520853046949206086, −3.67545219018025828928975276315, −3.66670558149615136296702413028, −3.61543251607167131174393284128, −3.47168358821679744245330674291, −3.27123356598973375090224146472, −3.25193154110124970374940889773, −3.22795458348806667062441990397, −3.11714483401427496046186403879, −3.05617074454605267141143474860, −2.65206064793386665519228713089, −2.52612581413467688495525786445, −2.47396548344403083270739589024, −2.45301456653380728466430925955, −2.26594885261052879556155197929, −2.19771689892736792368805855817, −2.07493227321099959362270016913, −1.97931020895195407192277886426, −1.40034957243540127299196596602, −1.23874593222495461665327511491, −1.23772403742308812856794753077, −1.16154582798994874537984161609, −1.15879844892058751619980085733, −1.15810023499494673607249330343, −0.915044466323914462691926285327, 0.915044466323914462691926285327, 1.15810023499494673607249330343, 1.15879844892058751619980085733, 1.16154582798994874537984161609, 1.23772403742308812856794753077, 1.23874593222495461665327511491, 1.40034957243540127299196596602, 1.97931020895195407192277886426, 2.07493227321099959362270016913, 2.19771689892736792368805855817, 2.26594885261052879556155197929, 2.45301456653380728466430925955, 2.47396548344403083270739589024, 2.52612581413467688495525786445, 2.65206064793386665519228713089, 3.05617074454605267141143474860, 3.11714483401427496046186403879, 3.22795458348806667062441990397, 3.25193154110124970374940889773, 3.27123356598973375090224146472, 3.47168358821679744245330674291, 3.61543251607167131174393284128, 3.66670558149615136296702413028, 3.67545219018025828928975276315, 3.74791741496520853046949206086

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

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