↑

Properties

Label 20-23e20-1.1-c0e10-0-0
Degree $20$
Conductor $1.716\times 10^{27}$
Sign $1$
Analytic cond. $1.64485\times 10^{-6}$
Root an. cond. $0.513814$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s + 13-s + 18-s − 25-s + 26-s + 29-s + 31-s + 36-s + 39-s + 41-s − 10·47-s − 49-s − 50-s + 52-s + 58-s − 2·59-s + 62-s + 71-s + 73-s − 75-s + 78-s + 82-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s + 13-s + 18-s − 25-s + 26-s + 29-s + 31-s + 36-s + 39-s + 41-s − 10·47-s − 49-s − 50-s + 52-s + 58-s − 2·59-s + 62-s + 71-s + 73-s − 75-s + 78-s + 82-s + ⋯

Functional equation

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

Invariants

Degree: \(20\)
Conductor: \(23^{20}\)
Sign: $1$
Analytic conductor: \(1.64485\times 10^{-6}\)
Root analytic conductor: \(0.513814\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 23^{20} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6830816301\)
\(L(\frac12)\) \(\approx\) \(0.6830816301\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
29 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
31 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
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 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
47 \( ( 1 + T + T^{2} )^{10} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
71 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
73 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.31346669602735780501556666432, −4.31122786587855878910505728878, −4.25544739428256422348195987102, −4.11610886288602929012346213287, −3.73719927636632843619940681746, −3.50030186170233522560942329026, −3.48145704318226877760266580882, −3.47912672643543325040992832221, −3.45507613431955823868264944697, −3.38771423182477079091132231744, −3.24282800077695358780020091063, −3.19890610202039675015855632780, −2.91565639763028195984950920428, −2.78351403199132428138375055544, −2.53146292675253785001350500437, −2.46210895350047538355852698316, −2.41459347363938636346834332744, −2.39033656988996553716550406438, −1.98832976613431978869891846814, −1.90660409659246277446888356648, −1.61711021586480188255228548776, −1.49090835484882860715703639907, −1.37860327792291508325272120362, −1.33377017719317223005252703081, −1.18544133226214076330003986811, 1.18544133226214076330003986811, 1.33377017719317223005252703081, 1.37860327792291508325272120362, 1.49090835484882860715703639907, 1.61711021586480188255228548776, 1.90660409659246277446888356648, 1.98832976613431978869891846814, 2.39033656988996553716550406438, 2.41459347363938636346834332744, 2.46210895350047538355852698316, 2.53146292675253785001350500437, 2.78351403199132428138375055544, 2.91565639763028195984950920428, 3.19890610202039675015855632780, 3.24282800077695358780020091063, 3.38771423182477079091132231744, 3.45507613431955823868264944697, 3.47912672643543325040992832221, 3.48145704318226877760266580882, 3.50030186170233522560942329026, 3.73719927636632843619940681746, 4.11610886288602929012346213287, 4.25544739428256422348195987102, 4.31122786587855878910505728878, 4.31346669602735780501556666432

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

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