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Properties

Label 32-728e16-1.1-c1e16-0-1
Degree 3232
Conductor 6.224×10456.224\times 10^{45}
Sign 11
Analytic cond. 1.70034×10121.70034\times 10^{12}
Root an. cond. 2.411032.41103
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 32·4-s + 96·8-s + 4·9-s + 264·16-s + 32·18-s + 672·32-s + 128·36-s + 1.53e3·64-s − 16·71-s + 384·72-s + 192·79-s + 8·81-s − 80·113-s + 127-s + 3.26e3·128-s + 131-s + 137-s + 139-s − 128·142-s + 1.05e3·144-s + 149-s + 151-s + 157-s + 1.53e3·158-s + 64·162-s + 163-s + ⋯
L(s)  = 1  + 5.65·2-s + 16·4-s + 33.9·8-s + 4/3·9-s + 66·16-s + 7.54·18-s + 118.·32-s + 64/3·36-s + 192·64-s − 1.89·71-s + 45.2·72-s + 21.6·79-s + 8/9·81-s − 7.52·113-s + 0.0887·127-s + 288.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·142-s + 88·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 122.·158-s + 5.02·162-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=((248⋅716⋅1316)s/2 ΓC(s)16 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((248⋅716⋅1316)s/2 ΓC(s+1/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 248⋅716⋅13162^{48} \cdot 7^{16} \cdot 13^{16}
Sign: 11
Analytic conductor: 1.70034×10121.70034\times 10^{12}
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 248⋅716⋅1316, ( :[1/2]16), 1)(32,\ 2^{48} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 552.9005622552.9005622
L(12)L(\frac12) ≈\approx 552.9005622552.9005622
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−pT+pT2−p2T3+p2T4)4 ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{4}
7 (1−p2T4+p4T8)2 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2}
13 1+36T2+648T4+11160T6+172319T8+11160p2T10+648p4T12+36p6T14+p8T16 1 + 36 T^{2} + 648 T^{4} + 11160 T^{6} + 172319 T^{8} + 11160 p^{2} T^{10} + 648 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16}
good3 (1−4T2+8T4−4p2T6+p4T8)2(1+4T2+8T4−40T6−161T8−40p2T10+8p4T12+4p6T14+p8T16) ( 1 - 4 T^{2} + 8 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 4 T^{2} + 8 T^{4} - 40 T^{6} - 161 T^{8} - 40 p^{2} T^{10} + 8 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} )
5 (1−12T2+72T4−264T6+959T8−264p2T10+72p4T12−12p6T14+p8T16)(1+12T2+72T4+264T6+959T8+264p2T10+72p4T12+12p6T14+p8T16) ( 1 - 12 T^{2} + 72 T^{4} - 264 T^{6} + 959 T^{8} - 264 p^{2} T^{10} + 72 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )( 1 + 12 T^{2} + 72 T^{4} + 264 T^{6} + 959 T^{8} + 264 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )
11 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
17 (1−pT2+p2T4)8 ( 1 - p T^{2} + p^{2} T^{4} )^{8}
19 (1−12T2+72T4+7800T6−177121T8+7800p2T10+72p4T12−12p6T14+p8T16)(1+12T2+72T4−7800T6−177121T8−7800p2T10+72p4T12+12p6T14+p8T16) ( 1 - 12 T^{2} + 72 T^{4} + 7800 T^{6} - 177121 T^{8} + 7800 p^{2} T^{10} + 72 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )( 1 + 12 T^{2} + 72 T^{4} - 7800 T^{6} - 177121 T^{8} - 7800 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )
23 (1−10T2+p2T4)4(1+10T2−429T4+10p2T6+p4T8)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{4}( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2}
29 (1+pT2+p2T4)8 ( 1 + p T^{2} + p^{2} T^{4} )^{8}
31 (1+p2T4)8 ( 1 + p^{2} T^{4} )^{8}
37 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
41 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
43 (1−pT2+p2T4)8 ( 1 - p T^{2} + p^{2} T^{4} )^{8}
47 (1+p2T4)8 ( 1 + p^{2} T^{4} )^{8}
53 (1−pT2)16 ( 1 - p T^{2} )^{16}
59 (1+108T2+5832T4+108p2T6+p4T8)2(1+108T2+5832T4−122040T6−18707521T8−122040p2T10+5832p4T12+108p6T14+p8T16) ( 1 + 108 T^{2} + 5832 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 108 T^{2} + 5832 T^{4} - 122040 T^{6} - 18707521 T^{8} - 122040 p^{2} T^{10} + 5832 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} )
61 (1−12T2+72T4−12p2T6+p4T8)2(1+12T2+72T4−88440T6−14376481T8−88440p2T10+72p4T12+12p6T14+p8T16) ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 12 T^{2} + 72 T^{4} - 88440 T^{6} - 14376481 T^{8} - 88440 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )
67 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
71 (1+8T+32T2+8pT3+p2T4)4(1−8T+32T2+880T3−8561T4+880pT5+32p2T6−8p3T7+p4T8)2 ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4}( 1 - 8 T + 32 T^{2} + 880 T^{3} - 8561 T^{4} + 880 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 (1+p2T4)8 ( 1 + p^{2} T^{4} )^{8}
79 (1−24T+288T2−24pT3+p2T4)8 ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{8}
83 (1−36T2+648T4−36p2T6+p4T8)2(1+36T2+648T4+36p2T6+p4T8)2 ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
97 (1−p2T4+p4T8)4 ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.88056533539562337084728601849, −2.61978261613170898514140458136, −2.56211113361121454795042050129, −2.52866567168379219296217075559, −2.51707975235063092831241712790, −2.46104452567267054667948867935, −2.33933484939565047274110697579, −2.21699020202223391859774242989, −2.10554860057620952021689434805, −2.08255833024506090130748978502, −2.05085024351402051369105920173, −1.94562581726464004814105325326, −1.86922413020738212092636753981, −1.63206469494266028296720689634, −1.59435055305456697360215051448, −1.53851866819779757178067921584, −1.46542677698757557217243930343, −1.24450174246982871107456969768, −1.17733725138867153865003491862, −1.09888698547469099368664323669, −0.865167441555352699809887864993, −0.865069511884623711385372939182, −0.854616048722037483270136764408, −0.60669728597602715207507623405, −0.14668513627062788227937934034, 0.14668513627062788227937934034, 0.60669728597602715207507623405, 0.854616048722037483270136764408, 0.865069511884623711385372939182, 0.865167441555352699809887864993, 1.09888698547469099368664323669, 1.17733725138867153865003491862, 1.24450174246982871107456969768, 1.46542677698757557217243930343, 1.53851866819779757178067921584, 1.59435055305456697360215051448, 1.63206469494266028296720689634, 1.86922413020738212092636753981, 1.94562581726464004814105325326, 2.05085024351402051369105920173, 2.08255833024506090130748978502, 2.10554860057620952021689434805, 2.21699020202223391859774242989, 2.33933484939565047274110697579, 2.46104452567267054667948867935, 2.51707975235063092831241712790, 2.52866567168379219296217075559, 2.56211113361121454795042050129, 2.61978261613170898514140458136, 2.88056533539562337084728601849

Graph of the ZZ-function along the critical line

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