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Properties

Label 48-2717e24-1.1-c0e24-0-1
Degree $48$
Conductor $2.619\times 10^{82}$
Sign $1$
Analytic cond. $1492.52$
Root an. cond. $1.16445$
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  − 3·2-s + 3·3-s + 6·4-s − 9·6-s − 8·8-s + 6·9-s + 12·11-s + 18·12-s + 9·16-s − 18·18-s − 36·22-s + 3·23-s − 24·24-s + 8·27-s − 6·32-s + 36·33-s + 36·36-s − 3·41-s + 72·44-s − 9·46-s + 27·48-s + 3·53-s − 24·54-s − 108·66-s + 9·69-s − 48·72-s + 9·81-s + ⋯
L(s)  = 1  − 3·2-s + 3·3-s + 6·4-s − 9·6-s − 8·8-s + 6·9-s + 12·11-s + 18·12-s + 9·16-s − 18·18-s − 36·22-s + 3·23-s − 24·24-s + 8·27-s − 6·32-s + 36·33-s + 36·36-s − 3·41-s + 72·44-s − 9·46-s + 27·48-s + 3·53-s − 24·54-s − 108·66-s + 9·69-s − 48·72-s + 9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{24} \cdot 13^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{24} \cdot 13^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(48\)
Conductor: \(11^{24} \cdot 13^{24} \cdot 19^{24}\)
Sign: $1$
Analytic conductor: \(1492.52\)
Root analytic conductor: \(1.16445\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((48,\ 11^{24} \cdot 13^{24} \cdot 19^{24} ,\ ( \ : [0]^{24} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(64.02383222\)
\(L(\frac12)\) \(\approx\) \(64.02383222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 - T + T^{2} )^{12} \)
13 \( ( 1 - T^{3} + T^{6} )^{4} \)
19 \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \)
good2 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \)
5 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
7 \( ( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} )^{2} \)
17 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \)
29 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
31 \( ( 1 - T + T^{2} )^{12}( 1 + T + T^{2} )^{12} \)
37 \( ( 1 - T )^{24}( 1 + T )^{24} \)
41 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \)
43 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
47 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \)
59 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
61 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
67 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
71 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
73 \( ( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
83 \( ( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} )^{2} \)
89 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
97 \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.83677945416074582568965599177, −1.76128826984647580236650158411, −1.74463996976702558433706954706, −1.70042152663876782877143037590, −1.61675823798770280219010353059, −1.56017488620558073024264239358, −1.46682941122912369337706308947, −1.40462800934705747054331333373, −1.38495178815640780141329103011, −1.37367954432414737424597964922, −1.33362841317408429065041531930, −1.30760507613651658153104125518, −1.25077857480098816280526102258, −1.11854225696499536659986067879, −1.09263303279911739270703398239, −1.08227980969358040978710317496, −1.02974141158706146493604301360, −1.01580357996056208996165149512, −0.943871871000069201144699546289, −0.866246506359621768826081250832, −0.857499514490901877232091431998, −0.75432382611091824128435719312, −0.74762704529894936654153735946, −0.69120150386144275968211190354, −0.49578995086526807509890301990, 0.49578995086526807509890301990, 0.69120150386144275968211190354, 0.74762704529894936654153735946, 0.75432382611091824128435719312, 0.857499514490901877232091431998, 0.866246506359621768826081250832, 0.943871871000069201144699546289, 1.01580357996056208996165149512, 1.02974141158706146493604301360, 1.08227980969358040978710317496, 1.09263303279911739270703398239, 1.11854225696499536659986067879, 1.25077857480098816280526102258, 1.30760507613651658153104125518, 1.33362841317408429065041531930, 1.37367954432414737424597964922, 1.38495178815640780141329103011, 1.40462800934705747054331333373, 1.46682941122912369337706308947, 1.56017488620558073024264239358, 1.61675823798770280219010353059, 1.70042152663876782877143037590, 1.74463996976702558433706954706, 1.76128826984647580236650158411, 1.83677945416074582568965599177

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

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