Properties

Label 40-404e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.342\times 10^{52}$
Sign $1$
Analytic cond. $1.23245\times 10^{-14}$
Root an. cond. $0.449023$
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  − 5·13-s − 10·17-s − 32-s − 5·61-s − 5·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·221-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 5·13-s − 10·17-s − 32-s − 5·61-s − 5·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 50·221-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 101^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 101^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{40} \cdot 101^{20}\)
Sign: $1$
Analytic conductor: \(1.23245\times 10^{-14}\)
Root analytic conductor: \(0.449023\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{404} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{40} \cdot 101^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007411272361\)
\(L(\frac12)\) \(\approx\) \(0.007411272361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
101 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
good3 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
5 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
7 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
11 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{10} \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
41 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
43 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
47 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
53 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
59 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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

−2.90988851759274814253596789412, −2.89064656875672964790906512934, −2.86116720594174462989549304218, −2.85356446163337558246027193514, −2.83036092763113532531951747623, −2.57495157628526989421961033027, −2.50593712585465730763233365284, −2.49576976475677993303283720544, −2.35699536337165054344293693292, −2.35680192243366114305958496144, −2.28410289493462730775820183978, −2.12865788153770767008274315388, −2.12855828021913603191146722350, −2.08708377605093776592735284629, −2.07572310612402283037005324535, −2.04627666451026924217809994555, −1.83708548387501276601184762813, −1.78528792672795120945510742226, −1.62026767266185514334382304882, −1.53361378080098745944304798823, −1.51516758300246311839125988170, −1.39312513326913642483377862286, −1.22195378149233920644209065716, −1.00930115635347560036163001700, −0.51712563085711301580363819109, 0.51712563085711301580363819109, 1.00930115635347560036163001700, 1.22195378149233920644209065716, 1.39312513326913642483377862286, 1.51516758300246311839125988170, 1.53361378080098745944304798823, 1.62026767266185514334382304882, 1.78528792672795120945510742226, 1.83708548387501276601184762813, 2.04627666451026924217809994555, 2.07572310612402283037005324535, 2.08708377605093776592735284629, 2.12855828021913603191146722350, 2.12865788153770767008274315388, 2.28410289493462730775820183978, 2.35680192243366114305958496144, 2.35699536337165054344293693292, 2.49576976475677993303283720544, 2.50593712585465730763233365284, 2.57495157628526989421961033027, 2.83036092763113532531951747623, 2.85356446163337558246027193514, 2.86116720594174462989549304218, 2.89064656875672964790906512934, 2.90988851759274814253596789412

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

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