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Properties

Label 20-993e10-1.1-c0e10-0-0
Degree $20$
Conductor $9.322\times 10^{29}$
Sign $1$
Analytic cond. $0.000893436$
Root an. cond. $0.703968$
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 + 10·4-s − 2·7-s − 10·12-s − 2·13-s + 55·16-s − 2·19-s + 2·21-s − 25-s − 20·28-s − 2·31-s − 2·37-s + 2·39-s − 2·43-s − 55·48-s + 49-s − 20·52-s + 2·57-s − 2·61-s + 220·64-s − 2·67-s − 2·73-s + 75-s − 20·76-s − 2·79-s + 20·84-s + 4·91-s + ⋯
L(s)  = 1  − 3-s + 10·4-s − 2·7-s − 10·12-s − 2·13-s + 55·16-s − 2·19-s + 2·21-s − 25-s − 20·28-s − 2·31-s − 2·37-s + 2·39-s − 2·43-s − 55·48-s + 49-s − 20·52-s + 2·57-s − 2·61-s + 220·64-s − 2·67-s − 2·73-s + 75-s − 20·76-s − 2·79-s + 20·84-s + 4·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 331^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 331^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{10} \cdot 331^{10}\)
Sign: $1$
Analytic conductor: \(0.000893436\)
Root analytic conductor: \(0.703968\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{10} \cdot 331^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.812595219\)
\(L(\frac12)\) \(\approx\) \(4.812595219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
331 \( ( 1 - T )^{10} \)
good2 \( ( 1 - T )^{10}( 1 + T )^{10} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
83 \( ( 1 - T )^{10}( 1 + T )^{10} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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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

−3.80162790320738436413546657156, −3.67855080632724925910887093158, −3.52240121918078791810403856848, −3.39305740814554794747371848048, −3.26166635931254772521913267968, −3.12664214682244725501775452235, −3.03572781129351431244183506272, −2.95154488288849071677486192750, −2.93885572902344142969213663256, −2.86170441120486204721578895912, −2.78608753000016962343141272713, −2.68194292764885798622996698068, −2.51754814313928536016244587205, −2.40509824697780296697175326497, −2.25377099444822116503320121368, −2.19178160930844498946317536912, −1.93353705460633757742405980599, −1.93013750144258147608592633519, −1.81948606889472962851987402389, −1.70159609296608369925323392651, −1.49128879098857467557410442995, −1.47644228651460761560375032749, −1.47286092203181215989423430899, −1.32318587584396066902072154877, −0.849648715295435619148643206514, 0.849648715295435619148643206514, 1.32318587584396066902072154877, 1.47286092203181215989423430899, 1.47644228651460761560375032749, 1.49128879098857467557410442995, 1.70159609296608369925323392651, 1.81948606889472962851987402389, 1.93013750144258147608592633519, 1.93353705460633757742405980599, 2.19178160930844498946317536912, 2.25377099444822116503320121368, 2.40509824697780296697175326497, 2.51754814313928536016244587205, 2.68194292764885798622996698068, 2.78608753000016962343141272713, 2.86170441120486204721578895912, 2.93885572902344142969213663256, 2.95154488288849071677486192750, 3.03572781129351431244183506272, 3.12664214682244725501775452235, 3.26166635931254772521913267968, 3.39305740814554794747371848048, 3.52240121918078791810403856848, 3.67855080632724925910887093158, 3.80162790320738436413546657156

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

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