Properties

Label 40-1359e20-1.1-c0e20-0-0
Degree $40$
Conductor $4.617\times 10^{62}$
Sign $1$
Analytic cond. $0.000424161$
Root an. cond. $0.823546$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·4-s + 10·16-s − 5·31-s − 5·37-s + 10·64-s + 5·103-s − 5·109-s − 25·124-s + 127-s + 131-s + 137-s + 139-s − 25·148-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 + ⋯
L(s)  = 1  + 5·4-s + 10·16-s − 5·31-s − 5·37-s + 10·64-s + 5·103-s − 5·109-s − 25·124-s + 127-s + 131-s + 137-s + 139-s − 25·148-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 151^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{40} \cdot 151^{20}\)
Sign: $1$
Analytic conductor: \(0.000424161\)
Root analytic conductor: \(0.823546\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1359} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{40} \cdot 151^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274399076\)
\(L(\frac12)\) \(\approx\) \(1.274399076\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
151 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
good2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
5 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
7 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
11 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
17 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
19 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
29 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
41 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
43 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \)
47 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
53 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \)
61 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
97 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.28159260847323008345206336077, −2.28155916331868119028007414349, −2.21320767726322418974504119334, −2.12922640596419630869940657949, −2.09882974744794936450499561840, −2.01144154386785541177519673621, −1.97282947933049369553553758835, −1.96598632983747408463116708417, −1.90123932635303870706050548562, −1.85919477169255140645299130696, −1.80698452316654623059925013661, −1.69124167382591436595166876114, −1.64029634497934845396570303070, −1.58615867058330829752218763986, −1.50034659661583888465841534006, −1.49561149640398963218925530278, −1.43746327434114551256849676678, −1.27816855089990146138542744716, −1.21681055315206616641695806663, −1.10068769251878645030433556137, −0.968393060239345825321019423699, −0.935818852222127903138219425790, −0.72799359552918239191804171769, −0.69377189933055202663425014010, −0.29219419956382558466585590418, 0.29219419956382558466585590418, 0.69377189933055202663425014010, 0.72799359552918239191804171769, 0.935818852222127903138219425790, 0.968393060239345825321019423699, 1.10068769251878645030433556137, 1.21681055315206616641695806663, 1.27816855089990146138542744716, 1.43746327434114551256849676678, 1.49561149640398963218925530278, 1.50034659661583888465841534006, 1.58615867058330829752218763986, 1.64029634497934845396570303070, 1.69124167382591436595166876114, 1.80698452316654623059925013661, 1.85919477169255140645299130696, 1.90123932635303870706050548562, 1.96598632983747408463116708417, 1.97282947933049369553553758835, 2.01144154386785541177519673621, 2.09882974744794936450499561840, 2.12922640596419630869940657949, 2.21320767726322418974504119334, 2.28155916331868119028007414349, 2.28159260847323008345206336077

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

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