# 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 of factors

## 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: 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

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1225355413$$ $$L(\frac12)$$ $$\approx$$ $$0.1225355413$$ $$L(1)$$ 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.