↑

Properties

Label 24-3248e12-1.1-c0e12-0-3
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  + 2·2-s + 4-s − 2·7-s + 2·9-s − 4·14-s + 4·18-s − 2·28-s − 2·29-s + 2·36-s − 4·43-s + 49-s − 2·53-s − 4·58-s − 4·63-s − 2·67-s − 12·79-s + 81-s − 8·86-s + 2·98-s − 4·106-s + 2·107-s − 2·109-s + 2·113-s − 2·116-s − 5·121-s − 8·126-s + 127-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·7-s + 2·9-s − 4·14-s + 4·18-s − 2·28-s − 2·29-s + 2·36-s − 4·43-s + 49-s − 2·53-s − 4·58-s − 4·63-s − 2·67-s − 12·79-s + 81-s − 8·86-s + 2·98-s − 4·106-s + 2·107-s − 2·109-s + 2·113-s − 2·116-s − 5·121-s − 8·126-s + 127-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 0.91871585850.9187158585
L(12)L(\frac12) ≈\approx 0.91871585850.9187158585
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−T+T2−T3+T4−T5+T6)2 ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}
7 (1+T+T2+T3+T4+T5+T6)2 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}
29 (1+T+T2+T3+T4+T5+T6)2 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}
good3 (1−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
5 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
11 (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} )
13 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
17 (1+T4)6 ( 1 + T^{4} )^{6}
19 (1−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
23 (1−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
31 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
37 (1−T2+T4−T6+T8−T10+T12)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}
41 (1+T4)6 ( 1 + T^{4} )^{6}
43 (1+T+T2+T3+T4+T5+T6)4 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}
47 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
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−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}
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−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
79 (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} )
83 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
89 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
97 1−T4+T8−T12+T16−T20+T24 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}
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

−2.99599150712857934941975770975, −2.96642885580976668833550417873, −2.90351166938755651144905297906, −2.73396214808628871645455393542, −2.48483352522857650389283157982, −2.44720181430312701323687841018, −2.32436560840621680276068853852, −2.27275103590952278915121683865, −2.19839818101697887045045046020, −2.05846576136210007949601661612, −1.92141718426481951010729312967, −1.88226202474570688424354551503, −1.87432723604898567349223508544, −1.74291324757815386648191074569, −1.58437005044490372033492446984, −1.50252191565474728465265066436, −1.41089130116961254677077454335, −1.36883316249414105390877704973, −1.34298662864208781908689187042, −1.14805958862100758529039845766, −1.11764691974148437954694957138, −0.962047704168188146380143389449, −0.55640568988175825485423691521, −0.34124707707942638013141617663, −0.21665812015984317353191543278, 0.21665812015984317353191543278, 0.34124707707942638013141617663, 0.55640568988175825485423691521, 0.962047704168188146380143389449, 1.11764691974148437954694957138, 1.14805958862100758529039845766, 1.34298662864208781908689187042, 1.36883316249414105390877704973, 1.41089130116961254677077454335, 1.50252191565474728465265066436, 1.58437005044490372033492446984, 1.74291324757815386648191074569, 1.87432723604898567349223508544, 1.88226202474570688424354551503, 1.92141718426481951010729312967, 2.05846576136210007949601661612, 2.19839818101697887045045046020, 2.27275103590952278915121683865, 2.32436560840621680276068853852, 2.44720181430312701323687841018, 2.48483352522857650389283157982, 2.73396214808628871645455393542, 2.90351166938755651144905297906, 2.96642885580976668833550417873, 2.99599150712857934941975770975

Graph of the ZZ-function along the critical line

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