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Properties

Label 20-644e10-1.1-c0e10-0-0
Degree $20$
Conductor $1.227\times 10^{28}$
Sign $1$
Analytic cond. $1.17606\times 10^{-5}$
Root an. cond. $0.566919$
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 + 7-s + 9-s − 2·11-s − 14-s − 18-s + 2·22-s + 23-s + 25-s + 2·29-s + 2·43-s − 46-s − 50-s − 2·58-s + 63-s − 2·67-s − 2·77-s − 2·79-s − 2·86-s − 2·99-s + 2·107-s − 11·109-s + 121-s − 126-s + 127-s + 131-s + 2·134-s + ⋯
L(s)  = 1  − 2-s + 7-s + 9-s − 2·11-s − 14-s − 18-s + 2·22-s + 23-s + 25-s + 2·29-s + 2·43-s − 46-s − 50-s − 2·58-s + 63-s − 2·67-s − 2·77-s − 2·79-s − 2·86-s − 2·99-s + 2·107-s − 11·109-s + 121-s − 126-s + 127-s + 131-s + 2·134-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 7^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(1.17606\times 10^{-5}\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1894609980\)
\(L(\frac12)\) \(\approx\) \(0.1894609980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
23 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
good3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
31 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
47 \( ( 1 + T^{2} )^{10} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
61 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
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

−4.14109622249776931501680517051, −4.06701421015025900571641161458, −3.91020713519631116002393765543, −3.81537901138590059114919552132, −3.74698686969377715478192995498, −3.69370958744439064356073814291, −3.65497160539527419559723338526, −3.37009940362141078419149190936, −3.26460258384793360858945680996, −2.88087373688414571315150999048, −2.86879480220487331381541703599, −2.76059965069675619935608786794, −2.70040520850210957600545557958, −2.68099531168247666146408690782, −2.66031742071056105082296293243, −2.42310386482465050497776102715, −2.27365781376578723709542965353, −2.12086698561808136966235554386, −1.74210575059876574475919925533, −1.61004919740896207520322029994, −1.48394625886130511999520856478, −1.41166859007223781262143576848, −1.24186542486451912069153592232, −1.05746836958542969239284626486, −0.74783838647248295121560878657, 0.74783838647248295121560878657, 1.05746836958542969239284626486, 1.24186542486451912069153592232, 1.41166859007223781262143576848, 1.48394625886130511999520856478, 1.61004919740896207520322029994, 1.74210575059876574475919925533, 2.12086698561808136966235554386, 2.27365781376578723709542965353, 2.42310386482465050497776102715, 2.66031742071056105082296293243, 2.68099531168247666146408690782, 2.70040520850210957600545557958, 2.76059965069675619935608786794, 2.86879480220487331381541703599, 2.88087373688414571315150999048, 3.26460258384793360858945680996, 3.37009940362141078419149190936, 3.65497160539527419559723338526, 3.69370958744439064356073814291, 3.74698686969377715478192995498, 3.81537901138590059114919552132, 3.91020713519631116002393765543, 4.06701421015025900571641161458, 4.14109622249776931501680517051

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

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