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Properties

Label 20-968e10-1.1-c0e10-0-0
Degree $20$
Conductor $7.224\times 10^{29}$
Sign $1$
Analytic cond. $0.000692348$
Root an. cond. $0.695050$
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 − 11-s − 2·17-s − 18-s − 2·19-s + 22-s − 25-s + 2·33-s + 2·34-s + 2·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s + 4·57-s − 2·59-s − 2·66-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 − 11-s − 2·17-s − 18-s − 2·19-s + 22-s − 25-s + 2·33-s + 2·34-s + 2·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s + 4·57-s − 2·59-s − 2·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s − 2·83-s + 2·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 11^{20}\)
Sign: $1$
Analytic conductor: \(0.000692348\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 11^{20} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003092656388\)
\(L(\frac12)\) \(\approx\) \(0.003092656388\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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

−3.89748358085256070358767441092, −3.68699942257320008663298287344, −3.67666754342356217576014390219, −3.66984984810837227081098562668, −3.49181965267738643326918394943, −3.45920759078992020957335703508, −3.26854252817840477687777295884, −3.20682747749628490461105744846, −2.79312644782053334533174752844, −2.78219615533764663470635608592, −2.75390114007876327385111713122, −2.67909177458267952115599807196, −2.60124744429732195254923037098, −2.51424441014398610289636369646, −2.23103163360888449882936653148, −1.95617721733854956908163359601, −1.89730209919324129188053565048, −1.79271957718723793427176027936, −1.76649340711532022859492725699, −1.58866435363166209748597817298, −1.44184266536347797154642462963, −1.27644318277454306596368813199, −1.01681605281555663097638216784, −0.45822483402786306813009836265, −0.10776527486643726426599983722, 0.10776527486643726426599983722, 0.45822483402786306813009836265, 1.01681605281555663097638216784, 1.27644318277454306596368813199, 1.44184266536347797154642462963, 1.58866435363166209748597817298, 1.76649340711532022859492725699, 1.79271957718723793427176027936, 1.89730209919324129188053565048, 1.95617721733854956908163359601, 2.23103163360888449882936653148, 2.51424441014398610289636369646, 2.60124744429732195254923037098, 2.67909177458267952115599807196, 2.75390114007876327385111713122, 2.78219615533764663470635608592, 2.79312644782053334533174752844, 3.20682747749628490461105744846, 3.26854252817840477687777295884, 3.45920759078992020957335703508, 3.49181965267738643326918394943, 3.66984984810837227081098562668, 3.67666754342356217576014390219, 3.68699942257320008663298287344, 3.89748358085256070358767441092

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

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