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Properties

Label 32-927e16-1.1-c0e16-0-0
Degree 3232
Conductor 2.974×10472.974\times 10^{47}
Sign 11
Analytic cond. 4.40341×10−64.40341\times 10^{-6}
Root an. cond. 0.6801710.680171
Motivic weight 00
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  + 4-s − 2·7-s + 2·13-s − 2·19-s − 25-s − 2·28-s + 49-s + 2·52-s + 2·61-s − 2·76-s + 2·79-s − 4·91-s − 15·97-s − 100-s + 103-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  + 4-s − 2·7-s + 2·13-s − 2·19-s − 25-s − 2·28-s + 49-s + 2·52-s + 2·61-s − 2·76-s + 2·79-s − 4·91-s − 15·97-s − 100-s + 103-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯

Functional equation

Λ(s)=((332⋅10316)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 103^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((332⋅10316)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 103^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 332⋅103163^{32} \cdot 103^{16}
Sign: 11
Analytic conductor: 4.40341×10−64.40341\times 10^{-6}
Root analytic conductor: 0.6801710.680171
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 332⋅10316, ( :[0]16), 1)(32,\ 3^{32} \cdot 103^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.35050029620.3505002962
L(12)L(\frac12) ≈\approx 0.35050029620.3505002962
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)
bad3 1 1
103 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16}
good2 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
5 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
7 (1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16)2 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{2}
11 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
13 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)2 ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{2}
17 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
19 (1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16)2 ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{2}
23 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
29 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
31 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
37 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
41 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
43 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
47 (1−T)16(1+T)16 ( 1 - T )^{16}( 1 + T )^{16}
53 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
59 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
61 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)2 ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{2}
67 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
71 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
73 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
79 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)2 ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{2}
83 1−T2+T4−T6+T8−T10+T12−T14+T16−T18+T20−T22+T24−T26+T28−T30+T32 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32}
89 (1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16)(1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10+T11+T12+T13+T14+T15+T16) ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )
97 (1+T)16(1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10−T11+T12−T13+T14−T15+T16) ( 1 + T )^{16}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )
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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

−2.93357805405892855154879204913, −2.89018719461219868752991161707, −2.82331257116847354831222449366, −2.71232868546548135029042947807, −2.60137446843220795654745994309, −2.47198849379318853711675659031, −2.46830257465253837004255807325, −2.40955946440900425819670650871, −2.34998374434137112277526819748, −2.33782942037391667626744387024, −2.03167008809096728319065461568, −2.01060155150798282923743455533, −1.95943751010902439467531085996, −1.89509596668334793348985322019, −1.73650375368086800303389775957, −1.58081593725565881571705110467, −1.53259855540372491047208685002, −1.51769311471812165477079952874, −1.47957576364833604445925150765, −1.41054860711271730454305805993, −1.28092566477451443098327616928, −0.980549395490329834323933257357, −0.836604648564770967463425890609, −0.69477744725725714660389238710, −0.59317343365352584630477050730, 0.59317343365352584630477050730, 0.69477744725725714660389238710, 0.836604648564770967463425890609, 0.980549395490329834323933257357, 1.28092566477451443098327616928, 1.41054860711271730454305805993, 1.47957576364833604445925150765, 1.51769311471812165477079952874, 1.53259855540372491047208685002, 1.58081593725565881571705110467, 1.73650375368086800303389775957, 1.89509596668334793348985322019, 1.95943751010902439467531085996, 2.01060155150798282923743455533, 2.03167008809096728319065461568, 2.33782942037391667626744387024, 2.34998374434137112277526819748, 2.40955946440900425819670650871, 2.46830257465253837004255807325, 2.47198849379318853711675659031, 2.60137446843220795654745994309, 2.71232868546548135029042947807, 2.82331257116847354831222449366, 2.89018719461219868752991161707, 2.93357805405892855154879204913

Graph of the ZZ-function along the critical line

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