Properties

Label 40-1397e20-1.1-c0e20-0-0
Degree $40$
Conductor $8.015\times 10^{62}$
Sign $1$
Analytic cond. $0.000736326$
Root an. cond. $0.834981$
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  − 5·9-s − 5·25-s − 2·32-s − 5·49-s + 10·81-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 + 25·225-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 5·9-s − 5·25-s − 2·32-s − 5·49-s + 10·81-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 + 25·225-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 127^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 127^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(11^{20} \cdot 127^{20}\)
Sign: $1$
Analytic conductor: \(0.000736326\)
Root analytic conductor: \(0.834981\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1397} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 11^{20} \cdot 127^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1242651807\)
\(L(\frac12)\) \(\approx\) \(0.1242651807\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
127 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
good2 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
3 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
13 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
17 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
19 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
23 \( ( 1 - T )^{20}( 1 + T )^{20} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
31 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
37 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
41 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
43 \( ( 1 - T )^{20}( 1 + T )^{20} \)
47 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
61 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
67 \( ( 1 - T )^{20}( 1 + T )^{20} \)
71 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
73 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
79 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
89 \( ( 1 - T )^{20}( 1 + T )^{20} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
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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

−2.40544277235678388703732477667, −2.30104076900515043261380218742, −2.17209863143811233063108689247, −2.16689610767286010915389261365, −2.14438922641477184279723965662, −2.11910562528975565077314174877, −2.08279506790874151940004932925, −1.94434819629363657821765438967, −1.74000522036254381379053028419, −1.67680548237131592296696975258, −1.66778008708919908239942021043, −1.66495682461118645321606326365, −1.61962578846122711736956143991, −1.52033901377789150030990754774, −1.42531575446534652501546038578, −1.40634619084195764566663201121, −1.39648918093132898370405970812, −1.33709805599424153400610138638, −1.13591963817489226489464160794, −1.10555176126170837305144471810, −0.71200173112291860965216860475, −0.68675180098394359613356607627, −0.46423514148740273336958818866, −0.44154511823781459240546527747, −0.39248191546864405841234853337, 0.39248191546864405841234853337, 0.44154511823781459240546527747, 0.46423514148740273336958818866, 0.68675180098394359613356607627, 0.71200173112291860965216860475, 1.10555176126170837305144471810, 1.13591963817489226489464160794, 1.33709805599424153400610138638, 1.39648918093132898370405970812, 1.40634619084195764566663201121, 1.42531575446534652501546038578, 1.52033901377789150030990754774, 1.61962578846122711736956143991, 1.66495682461118645321606326365, 1.66778008708919908239942021043, 1.67680548237131592296696975258, 1.74000522036254381379053028419, 1.94434819629363657821765438967, 2.08279506790874151940004932925, 2.11910562528975565077314174877, 2.14438922641477184279723965662, 2.16689610767286010915389261365, 2.17209863143811233063108689247, 2.30104076900515043261380218742, 2.40544277235678388703732477667

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

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