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Properties

Label 24-777e12-1.1-c0e12-0-0
Degree $24$
Conductor $4.842\times 10^{34}$
Sign $1$
Analytic cond. $1.15594\times 10^{-5}$
Root an. cond. $0.622714$
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  + 2·27-s − 6·37-s − 6·49-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 + 3·169-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 + ⋯
L(s)  = 1  + 2·27-s − 6·37-s − 6·49-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 + 3·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(1.15594\times 10^{-5}\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} \cdot 37^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2905842972\)
\(L(\frac12)\) \(\approx\) \(0.2905842972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T^{3} + T^{6} )^{2} \)
7 \( ( 1 + T^{2} )^{6} \)
37 \( ( 1 + T + T^{2} )^{6} \)
good2 \( 1 - T^{12} + T^{24} \)
5 \( 1 - T^{12} + T^{24} \)
11 \( ( 1 - T^{2} + T^{4} )^{6} \)
13 \( ( 1 - T^{2} + T^{4} )^{3}( 1 + T^{3} + T^{6} )^{2} \)
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 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
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 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
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

−3.69993767057856706391941981030, −3.27979261545093288326011352180, −3.26117625230773098495489005748, −3.14634836701883154426509508545, −3.12977911446331206300073129849, −3.12305463945194003421021207498, −3.11497062850005735141842848385, −3.04007743772798897095563976820, −3.01952257399358427857292061286, −2.97547351115251526570489235255, −2.59539168525026726646185688829, −2.53522248368411327413062242037, −2.31334958888635045957406470714, −2.16223302247262624370248328410, −2.03845118728657882622396008850, −2.03389226235948000895309686483, −1.83051021453559176929454149594, −1.78018618453124641901751376081, −1.71764397803064066375269774768, −1.61492011697014490754950302198, −1.54285392382399232254544733325, −1.21522264716944345645346632723, −1.05714443361472424776725058690, −1.00396668175066026783605755696, −0.54309993774226225227586438680, 0.54309993774226225227586438680, 1.00396668175066026783605755696, 1.05714443361472424776725058690, 1.21522264716944345645346632723, 1.54285392382399232254544733325, 1.61492011697014490754950302198, 1.71764397803064066375269774768, 1.78018618453124641901751376081, 1.83051021453559176929454149594, 2.03389226235948000895309686483, 2.03845118728657882622396008850, 2.16223302247262624370248328410, 2.31334958888635045957406470714, 2.53522248368411327413062242037, 2.59539168525026726646185688829, 2.97547351115251526570489235255, 3.01952257399358427857292061286, 3.04007743772798897095563976820, 3.11497062850005735141842848385, 3.12305463945194003421021207498, 3.12977911446331206300073129849, 3.14634836701883154426509508545, 3.26117625230773098495489005748, 3.27979261545093288326011352180, 3.69993767057856706391941981030

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

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