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Properties

Label 24-812e12-1.1-c0e12-0-1
Degree $24$
Conductor $8.216\times 10^{34}$
Sign $1$
Analytic cond. $1.96136\times 10^{-5}$
Root an. cond. $0.636585$
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  − 2·2-s + 4-s + 2·11-s − 4·22-s + 2·25-s + 2·37-s + 2·43-s + 2·44-s + 49-s − 4·50-s − 4·53-s + 4·71-s − 4·74-s − 2·79-s + 81-s − 4·86-s − 2·98-s + 2·100-s + 8·106-s − 12·113-s − 5·121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s − 8·142-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·11-s − 4·22-s + 2·25-s + 2·37-s + 2·43-s + 2·44-s + 49-s − 4·50-s − 4·53-s + 4·71-s − 4·74-s − 2·79-s + 81-s − 4·86-s − 2·98-s + 2·100-s + 8·106-s − 12·113-s − 5·121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s − 8·142-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 7^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(1.96136\times 10^{-5}\)
Root analytic conductor: \(0.636585\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1003332835\)
\(L(\frac12)\) \(\approx\) \(0.1003332835\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.61607576651119440833196890157, −3.56773592863378161282438772407, −3.52900369719269067588737905414, −3.34025302566291355020819175693, −3.25013561230798714620857304744, −2.87478224233576085726183889693, −2.81271668400874590573770523835, −2.72826485373890184525828278994, −2.62960042233324378593504830453, −2.61790387592244582965691064607, −2.60213823852317602143197242857, −2.57785687907230329299732356185, −2.53082997966486021647337749448, −2.51936981496033461507104549681, −1.92457804818515515731860166505, −1.91316523400475552723798879453, −1.71080187648303702300013194703, −1.70342969315409631318676179076, −1.50819218659160733738566878310, −1.49139679786644316045566473857, −1.24935009752536996031049085015, −1.08282476635178168899562458971, −1.03977187993041175460159173928, −1.01471141556400061288901033548, −0.53387397038050580295721247690, 0.53387397038050580295721247690, 1.01471141556400061288901033548, 1.03977187993041175460159173928, 1.08282476635178168899562458971, 1.24935009752536996031049085015, 1.49139679786644316045566473857, 1.50819218659160733738566878310, 1.70342969315409631318676179076, 1.71080187648303702300013194703, 1.91316523400475552723798879453, 1.92457804818515515731860166505, 2.51936981496033461507104549681, 2.53082997966486021647337749448, 2.57785687907230329299732356185, 2.60213823852317602143197242857, 2.61790387592244582965691064607, 2.62960042233324378593504830453, 2.72826485373890184525828278994, 2.81271668400874590573770523835, 2.87478224233576085726183889693, 3.25013561230798714620857304744, 3.34025302566291355020819175693, 3.52900369719269067588737905414, 3.56773592863378161282438772407, 3.61607576651119440833196890157

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

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