Properties

Label 40-1441e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.490\times 10^{63}$
Sign $1$
Analytic cond. $0.00136905$
Root an. cond. $0.848028$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·4-s + 10·16-s − 10·53-s − 10·64-s − 10·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·212-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 5·4-s + 10·16-s − 10·53-s − 10·64-s − 10·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·212-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 131^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 131^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(11^{20} \cdot 131^{20}\)
Sign: $1$
Analytic conductor: \(0.00136905\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 11^{20} \cdot 131^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001493381654\)
\(L(\frac12)\) \(\approx\) \(0.001493381654\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
131 \( ( 1 - T )^{20} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
3 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
5 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
7 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
13 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
23 \( ( 1 - T )^{20}( 1 + T )^{20} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
41 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
43 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{10} \)
59 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
61 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
67 \( ( 1 - T )^{20}( 1 + T )^{20} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{10} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.35352747654180101931699450028, −2.29300372664594227228657079201, −2.28482034253142266103178364332, −2.06135219026818999614812527308, −1.96735912859774328563135863310, −1.93637927423437068477113074040, −1.86122459171333006384042479221, −1.82389436877103448999361829499, −1.77950587063949820314605008006, −1.76504212503075879670923407166, −1.69996178684930393164765660168, −1.60724758205299805399170265567, −1.59425175736476051072354290947, −1.59040690498559854027892320784, −1.33723571415909356380006815584, −1.25191162432772337907878948151, −1.11721299176090846084419932670, −1.11117152733352728411108767987, −1.06009824413208863222803088997, −1.03024471473904774924360263067, −0.922909034634533040082382211927, −0.826779254878308605043999028751, −0.51579333842829241299304846383, −0.32571333603065004869830536085, −0.05563377149387688622689255639, 0.05563377149387688622689255639, 0.32571333603065004869830536085, 0.51579333842829241299304846383, 0.826779254878308605043999028751, 0.922909034634533040082382211927, 1.03024471473904774924360263067, 1.06009824413208863222803088997, 1.11117152733352728411108767987, 1.11721299176090846084419932670, 1.25191162432772337907878948151, 1.33723571415909356380006815584, 1.59040690498559854027892320784, 1.59425175736476051072354290947, 1.60724758205299805399170265567, 1.69996178684930393164765660168, 1.76504212503075879670923407166, 1.77950587063949820314605008006, 1.82389436877103448999361829499, 1.86122459171333006384042479221, 1.93637927423437068477113074040, 1.96735912859774328563135863310, 2.06135219026818999614812527308, 2.28482034253142266103178364332, 2.29300372664594227228657079201, 2.35352747654180101931699450028

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

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