Properties

Label 40-1375e20-1.1-c0e20-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.01950430293\)
\(L(\frac12)\) \(\approx\) \(0.01950430293\)
\(L(1)\) 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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
3 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
17 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
19 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
23 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
41 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
47 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
53 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \)
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 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
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

−2.46256272595576223407420774081, −2.23674837432607368230048325067, −2.23338645253155605948969509622, −2.20348861378972512932701772302, −2.08619148406347412890827850148, −2.06703270719170011411174762041, −2.04133171884518549813673489657, −1.84112987206505368754469864201, −1.74953871968524199025171124667, −1.61731323920266975435507739687, −1.54599592651136661385259086604, −1.53408678243934952776277110015, −1.50653561665538077210631884571, −1.47188784177027240342647154405, −1.47040419388496241360827868700, −1.46247967798656412897395712023, −1.45301004557504448097286585518, −1.40430755150021376517611223190, −0.906118023088098211926350677781, −0.898481222263506974973620214188, −0.883528886354449571503620346403, −0.76271225145232390612741148376, −0.72409773050268769876074204520, −0.28609424859910201703028926055, −0.19351589102559275432015516175, 0.19351589102559275432015516175, 0.28609424859910201703028926055, 0.72409773050268769876074204520, 0.76271225145232390612741148376, 0.883528886354449571503620346403, 0.898481222263506974973620214188, 0.906118023088098211926350677781, 1.40430755150021376517611223190, 1.45301004557504448097286585518, 1.46247967798656412897395712023, 1.47040419388496241360827868700, 1.47188784177027240342647154405, 1.50653561665538077210631884571, 1.53408678243934952776277110015, 1.54599592651136661385259086604, 1.61731323920266975435507739687, 1.74953871968524199025171124667, 1.84112987206505368754469864201, 2.04133171884518549813673489657, 2.06703270719170011411174762041, 2.08619148406347412890827850148, 2.20348861378972512932701772302, 2.23338645253155605948969509622, 2.23674837432607368230048325067, 2.46256272595576223407420774081

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

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