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Properties

Label 40-50e40-1.1-c0e20-0-0
Degree $40$
Conductor $9.095\times 10^{67}$
Sign $1$
Analytic cond. $83.5492$
Root an. cond. $1.11698$
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  + 32-s + 5·37-s − 5·49-s + 5·53-s − 5·89-s + 5·113-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 + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 32-s + 5·37-s − 5·49-s + 5·53-s − 5·89-s + 5·113-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 + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{80}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{80}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{40} \cdot 5^{80}\)
Sign: $1$
Analytic conductor: \(83.5492\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{40} \cdot 5^{80} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09823184524\)
\(L(\frac12)\) \(\approx\) \(0.09823184524\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{5} + T^{10} - T^{15} + T^{20} \)
5 \( 1 \)
good3 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
11 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
17 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
19 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
23 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
29 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
31 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 - T^{5} + T^{10} - T^{15} + T^{20} ) \)
41 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
47 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 - T^{5} + T^{10} - T^{15} + T^{20} ) \)
59 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
61 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
67 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
71 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
73 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
79 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
83 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
97 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
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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

−2.10058629419833295520733002239, −2.07338166036867033893925750324, −2.03974490488436002187741952532, −2.02422710735280483397205064845, −1.96370622630798282888357110793, −1.94127708886255272689682196813, −1.74031051747649616374192147916, −1.60884404948493290297849724190, −1.55620683160965715437764661181, −1.51817146244839601586808825022, −1.41196444717983609309731658127, −1.39638366299544826651566141186, −1.36824070591786412691067776774, −1.34390438524079270641693310894, −1.33205123081609099953144075107, −1.19155976233902373549987735947, −1.03831079036850339634791597982, −0.987321615158781961648542196360, −0.957359308938235313964507426259, −0.869689215502812323023211573453, −0.812039281499799658838052851464, −0.72796637263006638611037191383, −0.66838326453035904179235747731, −0.52598603765095327857967941628, −0.05211845347005572498188390321, 0.05211845347005572498188390321, 0.52598603765095327857967941628, 0.66838326453035904179235747731, 0.72796637263006638611037191383, 0.812039281499799658838052851464, 0.869689215502812323023211573453, 0.957359308938235313964507426259, 0.987321615158781961648542196360, 1.03831079036850339634791597982, 1.19155976233902373549987735947, 1.33205123081609099953144075107, 1.34390438524079270641693310894, 1.36824070591786412691067776774, 1.39638366299544826651566141186, 1.41196444717983609309731658127, 1.51817146244839601586808825022, 1.55620683160965715437764661181, 1.60884404948493290297849724190, 1.74031051747649616374192147916, 1.94127708886255272689682196813, 1.96370622630798282888357110793, 2.02422710735280483397205064845, 2.03974490488436002187741952532, 2.07338166036867033893925750324, 2.10058629419833295520733002239

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

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