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Properties

Label 40-1265e20-1.1-c0e20-0-0
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 − 2·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s − 2·53-s + 2·67-s + 4·69-s − 18·71-s − 2·75-s + 81-s + 2·97-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 + ⋯
L(s)  = 1  − 2·3-s + 2·9-s + 16-s − 2·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s − 2·53-s + 2·67-s + 4·69-s − 18·71-s − 2·75-s + 81-s + 2·97-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 + ⋯

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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 5^{20} \cdot 11^{20} \cdot 23^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1225355413\)
\(L(\frac12)\) \(\approx\) \(0.1225355413\)
\(L(1)\) 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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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 \( ( 1 + T^{2} )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.25247644711520867819422995478, −2.24746010450373425300338363097, −2.18882456922286416356834774633, −2.15073227690020646821258623075, −2.08847940842542297501755607174, −2.07295873061305068212832199125, −2.02920205789883530401391634685, −1.99158440774119638652257699992, −1.81128363330431324429098422154, −1.71264160137266787863906648386, −1.63423172127096425981967052543, −1.58898958651855492553034666464, −1.57176856165726017223677217405, −1.53605328183081804304882035269, −1.44386304369985993706027701968, −1.36613052924544473530824701388, −1.30810041960204307561733875000, −1.14203223939616845771320464952, −1.14018959798824026295549172815, −1.09584188699453839423952550851, −1.07011833281080179796373278089, −1.01485944148036493152162150235, −0.47288489317332877783183378660, −0.46600752894792214705177284237, −0.40200723529655240661595171774, 0.40200723529655240661595171774, 0.46600752894792214705177284237, 0.47288489317332877783183378660, 1.01485944148036493152162150235, 1.07011833281080179796373278089, 1.09584188699453839423952550851, 1.14018959798824026295549172815, 1.14203223939616845771320464952, 1.30810041960204307561733875000, 1.36613052924544473530824701388, 1.44386304369985993706027701968, 1.53605328183081804304882035269, 1.57176856165726017223677217405, 1.58898958651855492553034666464, 1.63423172127096425981967052543, 1.71264160137266787863906648386, 1.81128363330431324429098422154, 1.99158440774119638652257699992, 2.02920205789883530401391634685, 2.07295873061305068212832199125, 2.08847940842542297501755607174, 2.15073227690020646821258623075, 2.18882456922286416356834774633, 2.24746010450373425300338363097, 2.25247644711520867819422995478

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

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