↑

Properties

Label 24-3248e12-1.1-c0e12-0-7
Degree 2424
Conductor 1.378×10421.378\times 10^{42}
Sign 11
Analytic cond. 329.062329.062
Root an. cond. 1.273171.27317
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

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

Λ(s)=((248⋅712⋅2912)s/2 ΓC(s)12 L(s)=( Λ(1−s)\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}
Λ(s)=((248⋅712⋅2912)s/2 ΓC(s)12 L(s)=( Λ(1−s)\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: 2424
Conductor: 248⋅712⋅29122^{48} \cdot 7^{12} \cdot 29^{12}
Sign: 11
Analytic conductor: 329.062329.062
Root analytic conductor: 1.273171.27317
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 248⋅712⋅2912, ( :[0]12), 1)(24,\ 2^{48} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 17.6444307617.64443076
L(12)L(\frac12) ≈\approx 17.6444307617.64443076
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−T2+T4−T6+T8−T10+T12 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}
7 1−T2+T4−T6+T8−T10+T12 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}
29 1−T2+T4−T6+T8−T10+T12 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}
good3 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
5 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
11 (1−T)12(1−T2+T4−T6+T8−T10+T12) ( 1 - T )^{12}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )
13 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
17 (1+T2)12 ( 1 + T^{2} )^{12}
19 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
23 (1−T+T2−T3+T4−T5+T6)2(1+T+T2+T3+T4+T5+T6)2 ( 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−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
37 (1+T+T2+T3+T4+T5+T6)2(1−T2+T4−T6+T8−T10+T12) ( 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 ( 1 - T )^{12}( 1 + T )^{12}
43 (1+T+T2+T3+T4+T5+T6)2(1−T2+T4−T6+T8−T10+T12) ( 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−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
53 (1+T+T2+T3+T4+T5+T6)2(1−T2+T4−T6+T8−T10+T12) ( 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+T4)6 ( 1 + T^{4} )^{6}
61 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
67 (1−T+T2−T3+T4−T5+T6)2(1−T2+T4−T6+T8−T10+T12) ( 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−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
73 (1−T+T2−T3+T4−T5+T6)2(1+T+T2+T3+T4+T5+T6)2 ( 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+T2)6(1−T2+T4−T6+T8−T10+T12) ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )
83 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
89 (1−T+T2−T3+T4−T5+T6)2(1+T+T2+T3+T4+T5+T6)2 ( 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−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
show more
show less
   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(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 ZZ-function along the critical line

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