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Properties

Label 24-1805e12-1.1-c0e12-0-1
Degree $24$
Conductor $1.196\times 10^{39}$
Sign $1$
Analytic cond. $0.285503$
Root an. cond. $0.949111$
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  − 6·7-s + 18·49-s − 6·83-s + 6·121-s + 2·125-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 + 233-s + 239-s + ⋯
L(s)  = 1  − 6·7-s + 18·49-s − 6·83-s + 6·121-s + 2·125-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 + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{12} \cdot 19^{24}\)
Sign: $1$
Analytic conductor: \(0.285503\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{12} \cdot 19^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2109359997\)
\(L(\frac12)\) \(\approx\) \(0.2109359997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( 1 \)
good2 \( 1 - T^{12} + T^{24} \)
3 \( 1 - T^{12} + T^{24} \)
7 \( ( 1 + T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \)
11 \( ( 1 - T^{2} + T^{4} )^{6} \)
13 \( 1 - T^{12} + T^{24} \)
17 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
23 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
29 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} )^{6} \)
37 \( ( 1 + T^{4} )^{6} \)
41 \( ( 1 - T^{6} + T^{12} )^{2} \)
43 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
47 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
53 \( 1 - T^{12} + T^{24} \)
59 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T^{6} + T^{12} )^{2} \)
67 \( 1 - T^{12} + T^{24} \)
71 \( ( 1 - T^{6} + T^{12} )^{2} \)
73 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
79 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 + T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
97 \( 1 - T^{12} + T^{24} \)
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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

−3.12371103530665380156743823547, −3.09753847322977760940785880775, −2.90151973512612285096008805602, −2.88939666291090181038396089750, −2.86117705127522593967541693902, −2.75882583676348105402845747999, −2.66928166433120977831415710777, −2.43232088467403646826475310481, −2.42468792117617895470378742576, −2.38414033580703548605117444181, −2.33865530084199106035105353313, −2.24014108874784690126845988077, −2.13006170959535352366564909369, −1.92277811208098632257182543360, −1.91634048152943944396380247678, −1.54445756601223157737288484673, −1.53763349035060186651787064285, −1.49717131690372015279730099094, −1.39083430413202575321031520525, −1.15973290705251031567623264986, −1.06209563989439886169501730479, −0.981078919682957247687741209926, −0.53367304276813458563710460650, −0.52441140954466917392522906680, −0.34005321573832572035404083816, 0.34005321573832572035404083816, 0.52441140954466917392522906680, 0.53367304276813458563710460650, 0.981078919682957247687741209926, 1.06209563989439886169501730479, 1.15973290705251031567623264986, 1.39083430413202575321031520525, 1.49717131690372015279730099094, 1.53763349035060186651787064285, 1.54445756601223157737288484673, 1.91634048152943944396380247678, 1.92277811208098632257182543360, 2.13006170959535352366564909369, 2.24014108874784690126845988077, 2.33865530084199106035105353313, 2.38414033580703548605117444181, 2.42468792117617895470378742576, 2.43232088467403646826475310481, 2.66928166433120977831415710777, 2.75882583676348105402845747999, 2.86117705127522593967541693902, 2.88939666291090181038396089750, 2.90151973512612285096008805602, 3.09753847322977760940785880775, 3.12371103530665380156743823547

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

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