Properties

Label 40-1265e20-1.1-c0e20-0-1
Degree $40$
Conductor $1.101\times 10^{62}$
Sign $1$
Analytic cond. $0.000101146$
Root an. cond. $0.794554$
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  − 2·3-s + 2·9-s + 16-s + 25-s − 2·27-s − 2·37-s − 2·47-s − 2·48-s + 2·53-s − 2·67-s − 18·71-s − 2·75-s + 81-s + 20·97-s + 2·103-s + 4·111-s − 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 2·3-s + 2·9-s + 16-s + 25-s − 2·27-s − 2·37-s − 2·47-s − 2·48-s + 2·53-s − 2·67-s − 18·71-s − 2·75-s + 81-s + 20·97-s + 2·103-s + 4·111-s − 2·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + 157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(5^{20} \cdot 11^{20} \cdot 23^{20}\)
Sign: $1$
Analytic conductor: \(0.000101146\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1265} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 5^{20} \cdot 11^{20} \cdot 23^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2821495831\)
\(L(\frac12)\) \(\approx\) \(0.2821495831\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
17 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
43 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
71 \( ( 1 + T )^{20}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
97 \( ( 1 - T )^{20}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.49876678301790623389388227086, −2.31873928207491667087945879164, −2.08394210292917337806291909073, −2.04625133138882173199972418923, −2.02355256102648498403127785727, −1.97339262219989530286438261261, −1.97224138171408315703044642393, −1.93052585961352454196997767130, −1.90277443464937260409267725388, −1.72759748527625001754975551511, −1.71313274724946622317394512328, −1.68914521358438210380357112673, −1.60499862513657009616035627716, −1.49919902195778170443393597985, −1.49058253885776570308989091637, −1.44266706415850949693418859757, −1.29610639852663413189511207186, −1.25540942747564719508914616496, −1.21846583812730669002315519543, −0.884265203369929879497927880609, −0.878367033183943946794112739360, −0.789040340287500112945865925221, −0.64265148100214068703742333256, −0.57367391216193611081334421548, −0.54647939050369131070633939717, 0.54647939050369131070633939717, 0.57367391216193611081334421548, 0.64265148100214068703742333256, 0.789040340287500112945865925221, 0.878367033183943946794112739360, 0.884265203369929879497927880609, 1.21846583812730669002315519543, 1.25540942747564719508914616496, 1.29610639852663413189511207186, 1.44266706415850949693418859757, 1.49058253885776570308989091637, 1.49919902195778170443393597985, 1.60499862513657009616035627716, 1.68914521358438210380357112673, 1.71313274724946622317394512328, 1.72759748527625001754975551511, 1.90277443464937260409267725388, 1.93052585961352454196997767130, 1.97224138171408315703044642393, 1.97339262219989530286438261261, 2.02355256102648498403127785727, 2.04625133138882173199972418923, 2.08394210292917337806291909073, 2.31873928207491667087945879164, 2.49876678301790623389388227086

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

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