Properties

Label 40-500e20-1.1-c0e20-0-0
Degree $40$
Conductor $9.537\times 10^{53}$
Sign $1$
Analytic cond. $8.76077\times 10^{-13}$
Root an. cond. $0.499532$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{40} \cdot 5^{60}\)
Sign: $1$
Analytic conductor: \(8.76077\times 10^{-13}\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{40} \cdot 5^{60} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04121473787\)
\(L(\frac12)\) \(\approx\) \(0.04121473787\)
\(L(1)\) 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 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
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

−2.80558631459364303782571880132, −2.80092787215675148279023837085, −2.77526670832158424717307500651, −2.74844721050056203426963591745, −2.68747926537103619156871393896, −2.64140822529207079351385635157, −2.62524001360852216943637966430, −2.55255749585969842413646321168, −2.30116097857052367807428886908, −2.19630064763167192328056236597, −2.02917102699225512783803866495, −2.01781518817446259641593399392, −1.79165070802897530418121007781, −1.77834810822693312369575499424, −1.72366068682744459228268545979, −1.72276968112219109665431214603, −1.68248539810896953660076410039, −1.63327058147142198140840081932, −1.57861213360098084116305391312, −1.57786285725741538891877219249, −1.50190675476821892391134784656, −1.18729550736458619462429373059, −1.15237680994844736552034500079, −0.875338265131796993575081512371, −0.813853844280013572891209839038, 0.813853844280013572891209839038, 0.875338265131796993575081512371, 1.15237680994844736552034500079, 1.18729550736458619462429373059, 1.50190675476821892391134784656, 1.57786285725741538891877219249, 1.57861213360098084116305391312, 1.63327058147142198140840081932, 1.68248539810896953660076410039, 1.72276968112219109665431214603, 1.72366068682744459228268545979, 1.77834810822693312369575499424, 1.79165070802897530418121007781, 2.01781518817446259641593399392, 2.02917102699225512783803866495, 2.19630064763167192328056236597, 2.30116097857052367807428886908, 2.55255749585969842413646321168, 2.62524001360852216943637966430, 2.64140822529207079351385635157, 2.68747926537103619156871393896, 2.74844721050056203426963591745, 2.77526670832158424717307500651, 2.80092787215675148279023837085, 2.80558631459364303782571880132

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

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