Properties

Label 40-1375e20-1.1-c0e20-0-1
Degree $40$
Conductor $5.835\times 10^{62}$
Sign $1$
Analytic cond. $0.000536039$
Root an. cond. $0.828380$
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  + 5·5-s + 10·25-s + 5·49-s − 5·59-s + 5·67-s − 5·103-s + 10·125-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 + ⋯
L(s)  = 1  + 5·5-s + 10·25-s + 5·49-s − 5·59-s + 5·67-s − 5·103-s + 10·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(5^{60} \cdot 11^{20}\)
Sign: $1$
Analytic conductor: \(0.000536039\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1375} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 5^{60} \cdot 11^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.663118891\)
\(L(\frac12)\) \(\approx\) \(2.663118891\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
11 \( 1 - T^{5} + T^{10} - T^{15} + T^{20} \)
good2 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
3 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
13 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
17 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
31 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
37 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
41 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
47 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
53 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
61 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
71 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
73 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
79 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
83 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
89 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
97 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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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

−2.41287710661874078143059109565, −2.29732581777403436243202925782, −2.24707115476144043199740900202, −2.22874527982273044802854840468, −2.15697220720568276998716105515, −2.15289711649007428283095471935, −1.99275620702869840469499402642, −1.86313076305075860826946815005, −1.83028878945929911818838507639, −1.79505804774312712030162403029, −1.68999365872134201553695082455, −1.60944277667116818686008988413, −1.51951211433278030176650159843, −1.45775368225439779630557728360, −1.43857664123639013649010175916, −1.43138466614939977835735629254, −1.40422919559022059353558344525, −1.35781071947619600870961983824, −1.13214874854336410026036614170, −1.08230890728316459557230742704, −1.01026032627064722823017040234, −0.998537371765420855711265507666, −0.76528468281201838721013149177, −0.65010933790305929851375691188, −0.49724346505651285665227898390, 0.49724346505651285665227898390, 0.65010933790305929851375691188, 0.76528468281201838721013149177, 0.998537371765420855711265507666, 1.01026032627064722823017040234, 1.08230890728316459557230742704, 1.13214874854336410026036614170, 1.35781071947619600870961983824, 1.40422919559022059353558344525, 1.43138466614939977835735629254, 1.43857664123639013649010175916, 1.45775368225439779630557728360, 1.51951211433278030176650159843, 1.60944277667116818686008988413, 1.68999365872134201553695082455, 1.79505804774312712030162403029, 1.83028878945929911818838507639, 1.86313076305075860826946815005, 1.99275620702869840469499402642, 2.15289711649007428283095471935, 2.15697220720568276998716105515, 2.22874527982273044802854840468, 2.24707115476144043199740900202, 2.29732581777403436243202925782, 2.41287710661874078143059109565

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

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