↑

Properties

Label 24-220e12-1.1-c1e12-0-0
Degree 2424
Conductor 1.286×10281.286\times 10^{28}
Sign 11
Analytic cond. 863.768863.768
Root an. cond. 1.325401.32540
Motivic weight 11
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  + 12·5-s + 10·9-s + 16-s + 78·25-s + 4·37-s + 120·45-s − 46·49-s − 4·53-s + 4·64-s + 12·80-s + 33·81-s + 36·89-s − 8·97-s + 24·113-s − 10·121-s + 364·125-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯
L(s)  = 1  + 5.36·5-s + 10/3·9-s + 1/4·16-s + 78/5·25-s + 0.657·37-s + 17.8·45-s − 6.57·49-s − 0.549·53-s + 1/2·64-s + 1.34·80-s + 11/3·81-s + 3.81·89-s − 0.812·97-s + 2.25·113-s − 0.909·121-s + 32.5·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯

Functional equation

Λ(s)=((224⋅512⋅1112)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅512⋅1112)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 224⋅512⋅11122^{24} \cdot 5^{12} \cdot 11^{12}
Sign: 11
Analytic conductor: 863.768863.768
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 224⋅512⋅1112, ( :[1/2]12), 1)(24,\ 2^{24} \cdot 5^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 17.5246437117.52464371
L(12)L(\frac12) ≈\approx 17.5246437117.52464371
L(32)L(\frac{3}{2}) 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−T4−p2T6−p2T8+p6T12 1 - T^{4} - p^{2} T^{6} - p^{2} T^{8} + p^{6} T^{12}
5 (1−T)12 ( 1 - T )^{12}
11 1+10T2+71T4+1548T6+71p2T8+10p4T10+p6T12 1 + 10 T^{2} + 71 T^{4} + 1548 T^{6} + 71 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12}
good3 (1−5T2+7pT4−82T6+7p3T8−5p4T10+p6T12)2 ( 1 - 5 T^{2} + 7 p T^{4} - 82 T^{6} + 7 p^{3} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} )^{2}
7 (1+23T2+271T4+2162T6+271p2T8+23p4T10+p6T12)2 ( 1 + 23 T^{2} + 271 T^{4} + 2162 T^{6} + 271 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12} )^{2}
13 (1−36T2+887T4−13240T6+887p2T8−36p4T10+p6T12)2 ( 1 - 36 T^{2} + 887 T^{4} - 13240 T^{6} + 887 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} )^{2}
17 (1−63T2+1769T4−33526T6+1769p2T8−63p4T10+p6T12)2 ( 1 - 63 T^{2} + 1769 T^{4} - 33526 T^{6} + 1769 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2}
19 (1+5pT2+4039T4+98546T6+4039p2T8+5p5T10+p6T12)2 ( 1 + 5 p T^{2} + 4039 T^{4} + 98546 T^{6} + 4039 p^{2} T^{8} + 5 p^{5} T^{10} + p^{6} T^{12} )^{2}
23 (1−108T2+5423T4−159016T6+5423p2T8−108p4T10+p6T12)2 ( 1 - 108 T^{2} + 5423 T^{4} - 159016 T^{6} + 5423 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} )^{2}
29 (1−77T2+3035T4−89022T6+3035p2T8−77p4T10+p6T12)2 ( 1 - 77 T^{2} + 3035 T^{4} - 89022 T^{6} + 3035 p^{2} T^{8} - 77 p^{4} T^{10} + p^{6} T^{12} )^{2}
31 (1−135T2+8735T4−341554T6+8735p2T8−135p4T10+p6T12)2 ( 1 - 135 T^{2} + 8735 T^{4} - 341554 T^{6} + 8735 p^{2} T^{8} - 135 p^{4} T^{10} + p^{6} T^{12} )^{2}
37 (1−T+31T2−286T3+31pT4−p2T5+p3T6)4 ( 1 - T + 31 T^{2} - 286 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4}
41 (1−70T2+3007T4−116308T6+3007p2T8−70p4T10+p6T12)2 ( 1 - 70 T^{2} + 3007 T^{4} - 116308 T^{6} + 3007 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} )^{2}
43 (1+118T2+6727T4+284788T6+6727p2T8+118p4T10+p6T12)2 ( 1 + 118 T^{2} + 6727 T^{4} + 284788 T^{6} + 6727 p^{2} T^{8} + 118 p^{4} T^{10} + p^{6} T^{12} )^{2}
47 (1−12T2+2463T4−12904T6+2463p2T8−12p4T10+p6T12)2 ( 1 - 12 T^{2} + 2463 T^{4} - 12904 T^{6} + 2463 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2}
53 (1+T+107T2+230T3+107pT4+p2T5+p3T6)4 ( 1 + T + 107 T^{2} + 230 T^{3} + 107 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4}
59 (1−270T2+32807T4−2401732T6+32807p2T8−270p4T10+p6T12)2 ( 1 - 270 T^{2} + 32807 T^{4} - 2401732 T^{6} + 32807 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} )^{2}
61 (1−301T2+41283T4−3242254T6+41283p2T8−301p4T10+p6T12)2 ( 1 - 301 T^{2} + 41283 T^{4} - 3242254 T^{6} + 41283 p^{2} T^{8} - 301 p^{4} T^{10} + p^{6} T^{12} )^{2}
67 (1−104T2+9935T4−659616T6+9935p2T8−104p4T10+p6T12)2 ( 1 - 104 T^{2} + 9935 T^{4} - 659616 T^{6} + 9935 p^{2} T^{8} - 104 p^{4} T^{10} + p^{6} T^{12} )^{2}
71 (1−239T2+30959T4−2696130T6+30959p2T8−239p4T10+p6T12)2 ( 1 - 239 T^{2} + 30959 T^{4} - 2696130 T^{6} + 30959 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} )^{2}
73 (1+100T2+13455T4+959608T6+13455p2T8+100p4T10+p6T12)2 ( 1 + 100 T^{2} + 13455 T^{4} + 959608 T^{6} + 13455 p^{2} T^{8} + 100 p^{4} T^{10} + p^{6} T^{12} )^{2}
79 (1−146T2+20831T4−1830108T6+20831p2T8−146p4T10+p6T12)2 ( 1 - 146 T^{2} + 20831 T^{4} - 1830108 T^{6} + 20831 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} )^{2}
83 (1+234T2+29975T4+2694668T6+29975p2T8+234p4T10+p6T12)2 ( 1 + 234 T^{2} + 29975 T^{4} + 2694668 T^{6} + 29975 p^{2} T^{8} + 234 p^{4} T^{10} + p^{6} T^{12} )^{2}
89 (1−9T+165T2−938T3+165pT4−9p2T5+p3T6)4 ( 1 - 9 T + 165 T^{2} - 938 T^{3} + 165 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{4}
97 (1+2T−33T2−4T3−33pT4+2p2T5+p3T6)4 ( 1 + 2 T - 33 T^{2} - 4 T^{3} - 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4}
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

−4.27632374393016544779193315144, −4.11011158411645853064363928512, −4.06603420002457643688347955046, −4.05757895536378426861978937982, −3.87048125769078804390204009766, −3.75754033695638333511766303319, −3.63202919247051812961959745631, −3.26514572116430547240248648191, −3.25144682741088494546602358675, −3.12752877759680587859872222956, −2.99201337623790660238674931330, −2.94324033466690018803420983697, −2.88393691080233970314463087343, −2.46491672664236253472528501560, −2.33192419550725962867296083877, −2.30761548835383446729383481953, −2.28232913634462863517935612389, −1.90912497607685358913359579227, −1.74571274180383667961085625767, −1.71844605583125438996747489063, −1.69148623158413078306913117523, −1.34539865300933431253980709046, −1.30298746480378800197004827864, −1.25823738211269500436308085280, −0.808740271005341622398044069913, 0.808740271005341622398044069913, 1.25823738211269500436308085280, 1.30298746480378800197004827864, 1.34539865300933431253980709046, 1.69148623158413078306913117523, 1.71844605583125438996747489063, 1.74571274180383667961085625767, 1.90912497607685358913359579227, 2.28232913634462863517935612389, 2.30761548835383446729383481953, 2.33192419550725962867296083877, 2.46491672664236253472528501560, 2.88393691080233970314463087343, 2.94324033466690018803420983697, 2.99201337623790660238674931330, 3.12752877759680587859872222956, 3.25144682741088494546602358675, 3.26514572116430547240248648191, 3.63202919247051812961959745631, 3.75754033695638333511766303319, 3.87048125769078804390204009766, 4.05757895536378426861978937982, 4.06603420002457643688347955046, 4.11011158411645853064363928512, 4.27632374393016544779193315144

Graph of the ZZ-function along the critical line

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