Properties

Label 24-584e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.574\times 10^{33}$
Sign $1$
Analytic cond. $3.75698\times 10^{-7}$
Root an. cond. $0.539864$
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  + 2·8-s − 6·11-s − 12·17-s + 64-s + 6·83-s − 12·88-s − 6·107-s + 15·121-s + 127-s + 131-s − 24·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 72·187-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2·8-s − 6·11-s − 12·17-s + 64-s + 6·83-s − 12·88-s − 6·107-s + 15·121-s + 127-s + 131-s − 24·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 72·187-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 73^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 73^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 73^{12}\)
Sign: $1$
Analytic conductor: \(3.75698\times 10^{-7}\)
Root analytic conductor: \(0.539864\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{584} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 73^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01121011934\)
\(L(\frac12)\) \(\approx\) \(0.01121011934\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{3} + T^{6} )^{2} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
good3 \( ( 1 - T^{6} + T^{12} )^{2} \)
5 \( 1 - T^{12} + T^{24} \)
7 \( ( 1 - T^{4} + T^{8} )^{3} \)
11 \( ( 1 + T + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
13 \( 1 - T^{12} + T^{24} \)
17 \( ( 1 + T )^{12}( 1 - T^{2} + T^{4} )^{3} \)
19 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{6} + T^{12} )^{2} \)
29 \( 1 - T^{12} + T^{24} \)
31 \( 1 - T^{12} + T^{24} \)
37 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
47 \( 1 - T^{12} + T^{24} \)
53 \( 1 - T^{12} + T^{24} \)
59 \( ( 1 + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T^{6} + T^{12} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{6} + T^{12} )^{2} \)
83 \( ( 1 - T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \)
89 \( ( 1 - T^{6} + T^{12} )^{2} \)
97 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.89789528817575521907621497179, −3.89557144090216970998214950010, −3.55778072966141449311865717213, −3.52476172626561950033596263266, −3.48672278611203343679075365641, −3.19537173375670985820170803867, −3.07504747583185594106488356957, −3.07369913759216728367823830840, −2.87799379820011164745660261618, −2.62197009912479093490149244212, −2.59935552592736945338016651880, −2.45680036004303883882236710160, −2.42984307725362461995427995790, −2.42513359341239693146930374115, −2.32392155517377823734241505557, −2.30020466291185628530495571980, −2.26981491909066789471601495537, −2.02310861141720175282484050157, −1.97457729619915791480671410074, −1.68774679822086432203071395926, −1.65753390821380795831855856264, −1.61179148699471166822384950431, −1.22098856806020543632165184110, −0.836721072166803734430939491454, −0.16346702233063328706927632794, 0.16346702233063328706927632794, 0.836721072166803734430939491454, 1.22098856806020543632165184110, 1.61179148699471166822384950431, 1.65753390821380795831855856264, 1.68774679822086432203071395926, 1.97457729619915791480671410074, 2.02310861141720175282484050157, 2.26981491909066789471601495537, 2.30020466291185628530495571980, 2.32392155517377823734241505557, 2.42513359341239693146930374115, 2.42984307725362461995427995790, 2.45680036004303883882236710160, 2.59935552592736945338016651880, 2.62197009912479093490149244212, 2.87799379820011164745660261618, 3.07369913759216728367823830840, 3.07504747583185594106488356957, 3.19537173375670985820170803867, 3.48672278611203343679075365641, 3.52476172626561950033596263266, 3.55778072966141449311865717213, 3.89557144090216970998214950010, 3.89789528817575521907621497179

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

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