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Properties

Label 20-759e10-1.1-c0e10-0-0
Degree $20$
Conductor $6.345\times 10^{28}$
Sign $1$
Analytic cond. $6.08119\times 10^{-5}$
Root an. cond. $0.615459$
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  + 3-s + 4-s − 2·5-s + 11-s + 12-s − 2·15-s − 2·20-s − 23-s + 25-s − 2·31-s + 33-s + 44-s + 49-s + 2·53-s − 2·55-s − 2·60-s − 69-s − 11·71-s + 75-s + 2·89-s − 92-s − 2·93-s − 11·97-s + 100-s + 2·113-s + 2·115-s − 2·124-s + ⋯
L(s)  = 1  + 3-s + 4-s − 2·5-s + 11-s + 12-s − 2·15-s − 2·20-s − 23-s + 25-s − 2·31-s + 33-s + 44-s + 49-s + 2·53-s − 2·55-s − 2·60-s − 69-s − 11·71-s + 75-s + 2·89-s − 92-s − 2·93-s − 11·97-s + 100-s + 2·113-s + 2·115-s − 2·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{10} \cdot 11^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(6.08119\times 10^{-5}\)
Root analytic conductor: \(0.615459\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{10} \cdot 11^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4298626605\)
\(L(\frac12)\) \(\approx\) \(0.4298626605\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
11 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
good2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
19 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
61 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
71 \( ( 1 + T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
79 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
97 \( ( 1 + T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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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

−4.13169368230537108057237909656, −4.04512912232849494973551336004, −3.86755288993258236671664488487, −3.78012310341311797455699975717, −3.67290301885061116941650794845, −3.66244933132070070713906348165, −3.30176179402836884952348962350, −3.10427308500204328268688528516, −3.01682155691003008089464237517, −3.01559389182059657115658802879, −3.01332629440400930167828346057, −2.91570743693391176177554578745, −2.68216118391603710036992846260, −2.66414727933795613182352424142, −2.64245273271469038423740182094, −2.31996272233947653938819050435, −2.05180864387079366975673841630, −1.82068606928102639317806062344, −1.76889651018108094712603290338, −1.68069007868343073209400674611, −1.66851182074270606768332203879, −1.57218751463851208188347919730, −1.34855651537476016162739000380, −0.843907767160929969591719817391, −0.64511802082160709157692683767, 0.64511802082160709157692683767, 0.843907767160929969591719817391, 1.34855651537476016162739000380, 1.57218751463851208188347919730, 1.66851182074270606768332203879, 1.68069007868343073209400674611, 1.76889651018108094712603290338, 1.82068606928102639317806062344, 2.05180864387079366975673841630, 2.31996272233947653938819050435, 2.64245273271469038423740182094, 2.66414727933795613182352424142, 2.68216118391603710036992846260, 2.91570743693391176177554578745, 3.01332629440400930167828346057, 3.01559389182059657115658802879, 3.01682155691003008089464237517, 3.10427308500204328268688528516, 3.30176179402836884952348962350, 3.66244933132070070713906348165, 3.67290301885061116941650794845, 3.78012310341311797455699975717, 3.86755288993258236671664488487, 4.04512912232849494973551336004, 4.13169368230537108057237909656

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

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