# 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 of factors

## 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.