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Properties

Label 20-3864e10-1.1-c0e10-0-2
Degree $20$
Conductor $7.419\times 10^{35}$
Sign $1$
Analytic cond. $711.119$
Root an. cond. $1.38866$
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 + 3-s + 6-s − 7-s − 2·13-s − 14-s + 2·17-s − 21-s + 23-s − 25-s − 2·26-s + 2·29-s − 2·31-s + 2·34-s − 2·39-s − 9·41-s − 42-s + 9·43-s + 46-s − 50-s + 2·51-s + 2·53-s + 2·58-s − 9·59-s − 2·61-s − 2·62-s − 2·67-s + ⋯
L(s)  = 1  + 2-s + 3-s + 6-s − 7-s − 2·13-s − 14-s + 2·17-s − 21-s + 23-s − 25-s − 2·26-s + 2·29-s − 2·31-s + 2·34-s − 2·39-s − 9·41-s − 42-s + 9·43-s + 46-s − 50-s + 2·51-s + 2·53-s + 2·58-s − 9·59-s − 2·61-s − 2·62-s − 2·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(711.119\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.730395170\)
\(L(\frac12)\) \(\approx\) \(2.730395170\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
23 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
good5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} ) \)
43 \( ( 1 - T )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
47 \( ( 1 - T )^{10}( 1 + T )^{10} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
59 \( ( 1 + T )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
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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

−3.12578068713405995343580641640, −3.07214838311637436780246557086, −3.05056123472859725104801230654, −2.95214911705127405713504010768, −2.87320333135196834436784998666, −2.86130588713278802349047781569, −2.64292601468562311014210780838, −2.58597416174207090782499167980, −2.41037707069357776121431564119, −2.32754005477845840637814540909, −2.21791592566477313232789895415, −2.00985949013689289383204081199, −1.99036591580429104406534385724, −1.88288112473790507444242735772, −1.87955938062060410866079736869, −1.85541530902308369486591206524, −1.40837933421023581832410634056, −1.37388640867498040283553489101, −1.27693823761742975109907337757, −1.21666804948121053734413287137, −1.19831140607411464383741161286, −1.10456139279841582313102518272, −0.47301570533244034095877138987, −0.44653285970787922910637549761, −0.41785987883937374187561179182, 0.41785987883937374187561179182, 0.44653285970787922910637549761, 0.47301570533244034095877138987, 1.10456139279841582313102518272, 1.19831140607411464383741161286, 1.21666804948121053734413287137, 1.27693823761742975109907337757, 1.37388640867498040283553489101, 1.40837933421023581832410634056, 1.85541530902308369486591206524, 1.87955938062060410866079736869, 1.88288112473790507444242735772, 1.99036591580429104406534385724, 2.00985949013689289383204081199, 2.21791592566477313232789895415, 2.32754005477845840637814540909, 2.41037707069357776121431564119, 2.58597416174207090782499167980, 2.64292601468562311014210780838, 2.86130588713278802349047781569, 2.87320333135196834436784998666, 2.95214911705127405713504010768, 3.05056123472859725104801230654, 3.07214838311637436780246557086, 3.12578068713405995343580641640

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

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