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Properties

Label 24-2001e12-1.1-c0e12-0-0
Degree $24$
Conductor $4.121\times 10^{39}$
Sign $1$
Analytic cond. $0.983672$
Root an. cond. $0.999314$
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 − 5·4-s + 12·8-s + 9-s + 15·16-s − 2·18-s − 2·25-s + 2·31-s − 42·32-s − 5·36-s + 2·41-s − 2·47-s − 2·49-s + 4·50-s − 4·62-s − 35·64-s + 12·72-s + 2·73-s − 4·82-s + 4·94-s + 4·98-s + 10·100-s + 2·101-s − 10·124-s + 127-s + 114·128-s + 131-s + ⋯
L(s)  = 1  − 2·2-s − 5·4-s + 12·8-s + 9-s + 15·16-s − 2·18-s − 2·25-s + 2·31-s − 42·32-s − 5·36-s + 2·41-s − 2·47-s − 2·49-s + 4·50-s − 4·62-s − 35·64-s + 12·72-s + 2·73-s − 4·82-s + 4·94-s + 4·98-s + 10·100-s + 2·101-s − 10·124-s + 127-s + 114·128-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 23^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(0.983672\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 23^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0005520689526\)
\(L(\frac12)\) \(\approx\) \(0.0005520689526\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good2 \( ( 1 + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
17 \( ( 1 + T^{4} )^{6} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
37 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
79 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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

−3.00070842589473426903744912856, −2.91094564209163515288697300508, −2.81856833613691042997145679893, −2.81714908532823150658520983667, −2.78416199230669708858833641516, −2.75443183476180981967800331239, −2.53677183602777746486504616259, −2.51372762243530060108187544018, −2.32180074601726344856533534853, −2.21900001978340163416269799817, −2.10379207565987332566114184292, −2.10026283813814153116905633521, −1.85172201752438169174746694581, −1.70016150924670841522745612004, −1.52119896990927100255868071971, −1.38548675979364395157753701400, −1.30442746908178771324491254731, −1.28022904199962708516313214069, −1.21559059870066960333625837851, −1.05559255762933578757781712889, −1.04591744591398606015232252720, −0.990033349502577949822383483533, −0.66751816489813464279854681340, −0.21802360695063940453925465212, −0.06571674185132468088912579087, 0.06571674185132468088912579087, 0.21802360695063940453925465212, 0.66751816489813464279854681340, 0.990033349502577949822383483533, 1.04591744591398606015232252720, 1.05559255762933578757781712889, 1.21559059870066960333625837851, 1.28022904199962708516313214069, 1.30442746908178771324491254731, 1.38548675979364395157753701400, 1.52119896990927100255868071971, 1.70016150924670841522745612004, 1.85172201752438169174746694581, 2.10026283813814153116905633521, 2.10379207565987332566114184292, 2.21900001978340163416269799817, 2.32180074601726344856533534853, 2.51372762243530060108187544018, 2.53677183602777746486504616259, 2.75443183476180981967800331239, 2.78416199230669708858833641516, 2.81714908532823150658520983667, 2.81856833613691042997145679893, 2.91094564209163515288697300508, 3.00070842589473426903744912856

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

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