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Properties

Label 40-453e20-1.1-c0e20-0-0
Degree $40$
Conductor $1.324\times 10^{53}$
Sign $1$
Analytic cond. $1.21648\times 10^{-13}$
Root an. cond. $0.475474$
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  − 5·4-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

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

Invariants

Degree: \(40\)
Conductor: \(3^{20} \cdot 151^{20}\)
Sign: $1$
Analytic conductor: \(1.21648\times 10^{-13}\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{20} \cdot 151^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001394300122\)
\(L(\frac12)\) \(\approx\) \(0.001394300122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
151 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
5 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
7 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
11 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
13 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
17 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
53 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
61 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
67 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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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

−2.99244203819026722962921975877, −2.92887208714141529370633402191, −2.89989666068012529645773382489, −2.88342873760545828182090348197, −2.54685136610905343252326267959, −2.52841782061051431094780774400, −2.41166692791553024309080987588, −2.38464112699019949780195248097, −2.38207654027055214659021692046, −2.37215847792831890311071634457, −2.19317784519943015096626756890, −2.16382140834274056598001907531, −2.11402466544849394990064177644, −2.01181561364232635277688278676, −1.91681145185990929842250150677, −1.70480447967165128942087568446, −1.59654921021572787441263118420, −1.54448663124299700064814488540, −1.43317126145868440553404310603, −1.28928219755368866819241527817, −1.25200924385254999538081525569, −1.23788316917935426543115565974, −1.16839593808152149693475151303, −1.14210922338534384048802664827, −0.22217067236388223334165070435, 0.22217067236388223334165070435, 1.14210922338534384048802664827, 1.16839593808152149693475151303, 1.23788316917935426543115565974, 1.25200924385254999538081525569, 1.28928219755368866819241527817, 1.43317126145868440553404310603, 1.54448663124299700064814488540, 1.59654921021572787441263118420, 1.70480447967165128942087568446, 1.91681145185990929842250150677, 2.01181561364232635277688278676, 2.11402466544849394990064177644, 2.16382140834274056598001907531, 2.19317784519943015096626756890, 2.37215847792831890311071634457, 2.38207654027055214659021692046, 2.38464112699019949780195248097, 2.41166692791553024309080987588, 2.52841782061051431094780774400, 2.54685136610905343252326267959, 2.88342873760545828182090348197, 2.89989666068012529645773382489, 2.92887208714141529370633402191, 2.99244203819026722962921975877

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

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