Properties

Label 40-712e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.121\times 10^{57}$
Sign $1$
Analytic cond. $1.02977\times 10^{-9}$
Root an. cond. $0.596099$
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  + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·9-s − 2·12-s + 4·18-s − 2·19-s + 2·25-s − 2·27-s + 2·36-s − 4·38-s + 2·41-s + 2·43-s + 4·50-s − 4·54-s + 4·57-s − 2·59-s − 4·75-s − 2·76-s + 81-s + 4·82-s + 2·83-s + 4·86-s + 2·89-s − 4·97-s + 2·100-s + ⋯
L(s)  = 1  + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·9-s − 2·12-s + 4·18-s − 2·19-s + 2·25-s − 2·27-s + 2·36-s − 4·38-s + 2·41-s + 2·43-s + 4·50-s − 4·54-s + 4·57-s − 2·59-s − 4·75-s − 2·76-s + 81-s + 4·82-s + 2·83-s + 4·86-s + 2·89-s − 4·97-s + 2·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{60} \cdot 89^{20}\)
Sign: $1$
Analytic conductor: \(1.02977\times 10^{-9}\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{712} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{60} \cdot 89^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1595745096\)
\(L(\frac12)\) \(\approx\) \(0.1595745096\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
good3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
11 \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
17 \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
29 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
31 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
37 \( ( 1 + T^{4} )^{10} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
61 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
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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

−2.67780527048255904571806746803, −2.64240277343721040094180467435, −2.47807395129496890565234716984, −2.46265067207093483053294352210, −2.44044882206428685175121067365, −2.27520708070380661986929873867, −2.26117913916977559249081262239, −2.20654960844176425300557546976, −2.19741629301470957274176104910, −2.19534363424892827770481020801, −2.19075390156466861190449203799, −2.04731181142751525673010614761, −1.87780954807291510019875714084, −1.61745954927173754872982462118, −1.56677321841894246317455430573, −1.53958419053261519134445723691, −1.45395325214318122294701810424, −1.39761636828549573603735685266, −1.28365033677039488142098374428, −1.28094185708622593531553461456, −1.15124353710273176668984306261, −1.06365130455925400985305775541, −1.02009823127570854373240134823, −0.802986117586877474710365817099, −0.60152963256656413118709655715, 0.60152963256656413118709655715, 0.802986117586877474710365817099, 1.02009823127570854373240134823, 1.06365130455925400985305775541, 1.15124353710273176668984306261, 1.28094185708622593531553461456, 1.28365033677039488142098374428, 1.39761636828549573603735685266, 1.45395325214318122294701810424, 1.53958419053261519134445723691, 1.56677321841894246317455430573, 1.61745954927173754872982462118, 1.87780954807291510019875714084, 2.04731181142751525673010614761, 2.19075390156466861190449203799, 2.19534363424892827770481020801, 2.19741629301470957274176104910, 2.20654960844176425300557546976, 2.26117913916977559249081262239, 2.27520708070380661986929873867, 2.44044882206428685175121067365, 2.46265067207093483053294352210, 2.47807395129496890565234716984, 2.64240277343721040094180467435, 2.67780527048255904571806746803

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

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