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Properties

Label 24-580e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.449\times 10^{33}$
Sign $1$
Analytic cond. $3.45956\times 10^{-7}$
Root an. cond. $0.538012$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 2·9-s + 2·13-s − 4·18-s + 25-s − 4·26-s + 2·36-s − 2·41-s − 2·50-s + 2·52-s + 2·53-s + 2·61-s − 10·73-s + 81-s + 4·82-s − 2·89-s − 14·97-s + 100-s + 2·101-s − 4·106-s − 4·113-s + 4·117-s − 4·122-s + 127-s + 2·128-s + 131-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·9-s + 2·13-s − 4·18-s + 25-s − 4·26-s + 2·36-s − 2·41-s − 2·50-s + 2·52-s + 2·53-s + 2·61-s − 10·73-s + 81-s + 4·82-s − 2·89-s − 14·97-s + 100-s + 2·101-s − 4·106-s − 4·113-s + 4·117-s − 4·122-s + 127-s + 2·128-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(3.45956\times 10^{-7}\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06016623932\)
\(L(\frac12)\) \(\approx\) \(0.06016623932\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
31 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
53 \( ( 1 + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
59 \( ( 1 + T^{2} )^{12} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
67 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( ( 1 + T )^{12}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
79 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
97 \( ( 1 + T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−3.83313855384540800482942585486, −3.73760479114338860007969245564, −3.64511257994978781940458063528, −3.63836908977090793212742390287, −3.50927359691784838448583070470, −3.23970026358081480233577967147, −3.02854599929734781904987499141, −3.00873043791276160909398907779, −2.89878238689405844323246910363, −2.80725102044604023012291095808, −2.75062123749709962093078165071, −2.63904242292705613475051831101, −2.61365083104518836960731173802, −2.59708971688515454977857031413, −2.38262773568938247456531005224, −1.96869760703641494074346556243, −1.83241662952120390416440362411, −1.78752101784372838563881709381, −1.56276476817071677974681568973, −1.52020202574386384643122768923, −1.35795921922294321547072713129, −1.28913623869068778844719786976, −1.27142171318808300085702652794, −1.26867909280899533933080137707, −0.61338527950195630376983616365, 0.61338527950195630376983616365, 1.26867909280899533933080137707, 1.27142171318808300085702652794, 1.28913623869068778844719786976, 1.35795921922294321547072713129, 1.52020202574386384643122768923, 1.56276476817071677974681568973, 1.78752101784372838563881709381, 1.83241662952120390416440362411, 1.96869760703641494074346556243, 2.38262773568938247456531005224, 2.59708971688515454977857031413, 2.61365083104518836960731173802, 2.63904242292705613475051831101, 2.75062123749709962093078165071, 2.80725102044604023012291095808, 2.89878238689405844323246910363, 3.00873043791276160909398907779, 3.02854599929734781904987499141, 3.23970026358081480233577967147, 3.50927359691784838448583070470, 3.63836908977090793212742390287, 3.64511257994978781940458063528, 3.73760479114338860007969245564, 3.83313855384540800482942585486

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

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