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Properties

Label 24-3248e12-1.1-c0e12-0-7
Degree $24$
Conductor $1.378\times 10^{42}$
Sign $1$
Analytic cond. $329.062$
Root an. cond. $1.27317$
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  + 4-s + 12·11-s − 2·37-s − 2·43-s + 12·44-s + 49-s − 2·53-s + 2·67-s + 81-s − 2·107-s + 2·109-s + 79·121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4-s + 12·11-s − 2·37-s − 2·43-s + 12·44-s + 49-s − 2·53-s + 2·67-s + 81-s − 2·107-s + 2·109-s + 79·121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 7^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(329.062\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(17.64443076\)
\(L(\frac12)\) \(\approx\) \(17.64443076\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.95843389134644234697899628847, −2.93608350603408175364746341854, −2.75435547670834336779013510928, −2.54114596115368343624055619277, −2.36256425696156085571577316353, −2.28112793613820297990934497371, −2.15941836979808325829049342630, −2.10059898760082791508195332644, −2.06768912069511677101203908992, −2.00929761403908340850236482377, −1.98365648997287828679144797038, −1.88536069363254025627976705288, −1.78083527298022941358879775552, −1.77540040387310641676966908999, −1.57110004226724267899513218697, −1.40191099098740854169594123910, −1.37632440056115271320471939151, −1.32291368797986801770433195233, −1.14054741556959232903379953450, −1.13134420487580328222448579482, −1.11685776129584361277920743492, −0.951982011975907171926358901236, −0.890824938118078254450879360609, −0.853438179830430710299406696120, −0.47122874041287708120948129739, 0.47122874041287708120948129739, 0.853438179830430710299406696120, 0.890824938118078254450879360609, 0.951982011975907171926358901236, 1.11685776129584361277920743492, 1.13134420487580328222448579482, 1.14054741556959232903379953450, 1.32291368797986801770433195233, 1.37632440056115271320471939151, 1.40191099098740854169594123910, 1.57110004226724267899513218697, 1.77540040387310641676966908999, 1.78083527298022941358879775552, 1.88536069363254025627976705288, 1.98365648997287828679144797038, 2.00929761403908340850236482377, 2.06768912069511677101203908992, 2.10059898760082791508195332644, 2.15941836979808325829049342630, 2.28112793613820297990934497371, 2.36256425696156085571577316353, 2.54114596115368343624055619277, 2.75435547670834336779013510928, 2.93608350603408175364746341854, 2.95843389134644234697899628847

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

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