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Properties

Label 24-1332e12-1.1-c0e12-0-1
Degree $24$
Conductor $3.119\times 10^{37}$
Sign $1$
Analytic cond. $0.00744622$
Root an. cond. $0.815324$
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  + 6·37-s + 3·49-s − 6·103-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 6·37-s + 3·49-s − 6·103-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(0.00744622\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{24} \cdot 37^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6903641746\)
\(L(\frac12)\) \(\approx\) \(0.6903641746\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( ( 1 - T + T^{2} )^{6} \)
good5 \( 1 - T^{12} + T^{24} \)
7 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
11 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
13 \( ( 1 + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
17 \( 1 - T^{12} + T^{24} \)
19 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
23 \( ( 1 - T^{4} + T^{8} )^{3} \)
29 \( ( 1 - T^{4} + T^{8} )^{3} \)
31 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
41 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
47 \( ( 1 - T^{2} + T^{4} )^{6} \)
53 \( ( 1 - T^{6} + T^{12} )^{2} \)
59 \( 1 - T^{12} + T^{24} \)
61 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
67 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
71 \( ( 1 - T^{6} + T^{12} )^{2} \)
73 \( ( 1 - T^{6} + T^{12} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{6} + T^{12} )^{2} \)
89 \( 1 - T^{12} + T^{24} \)
97 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
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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

−3.32841181680964320142224985181, −3.12801713952591114319004930112, −2.92100633790253789288313397730, −2.77552250450566858781804456900, −2.76961957613150666642015515371, −2.73780296980753059833261498769, −2.70439616070906673974128368198, −2.58114987280814233221651109172, −2.52521572890585504187625238385, −2.51177254433627628489867784321, −2.49627958687087473392225619747, −2.47150732919114263874680433570, −2.43410113858641946287121224221, −2.00522407425586386852916414988, −1.89705613340982469175214563349, −1.72905186129401859337706077276, −1.67201343267568693463954039962, −1.38234501968309586224638023755, −1.38052324666097510762667824290, −1.31943291980369880571002887943, −1.24084188896274624280642742284, −1.14295224621796974807236369233, −1.02427042327457541716754468361, −0.813479760721868280176754139011, −0.39427750236906976356026398143, 0.39427750236906976356026398143, 0.813479760721868280176754139011, 1.02427042327457541716754468361, 1.14295224621796974807236369233, 1.24084188896274624280642742284, 1.31943291980369880571002887943, 1.38052324666097510762667824290, 1.38234501968309586224638023755, 1.67201343267568693463954039962, 1.72905186129401859337706077276, 1.89705613340982469175214563349, 2.00522407425586386852916414988, 2.43410113858641946287121224221, 2.47150732919114263874680433570, 2.49627958687087473392225619747, 2.51177254433627628489867784321, 2.52521572890585504187625238385, 2.58114987280814233221651109172, 2.70439616070906673974128368198, 2.73780296980753059833261498769, 2.76961957613150666642015515371, 2.77552250450566858781804456900, 2.92100633790253789288313397730, 3.12801713952591114319004930112, 3.32841181680964320142224985181

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

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