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Properties

Label 32-959e16-1.1-c0e16-0-0
Degree 3232
Conductor 5.118×10475.118\times 10^{47}
Sign 11
Analytic cond. 7.57899×10−67.57899\times 10^{-6}
Root an. cond. 0.6918110.691811
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  − 2·2-s + 4-s + 7-s + 9-s + 2·11-s − 2·14-s − 2·18-s − 4·22-s + 17·23-s + 25-s + 28-s + 36-s − 2·37-s + 2·44-s − 34·46-s − 2·50-s + 63-s − 17·71-s + 4·74-s + 2·77-s + 17·92-s + 2·99-s + 100-s + 2·107-s + 2·109-s + 121-s − 2·126-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 7-s + 9-s + 2·11-s − 2·14-s − 2·18-s − 4·22-s + 17·23-s + 25-s + 28-s + 36-s − 2·37-s + 2·44-s − 34·46-s − 2·50-s + 63-s − 17·71-s + 4·74-s + 2·77-s + 17·92-s + 2·99-s + 100-s + 2·107-s + 2·109-s + 121-s − 2·126-s + ⋯

Functional equation

Λ(s)=((716⋅13716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 137^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((716⋅13716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 137^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 716⋅137167^{16} \cdot 137^{16}
Sign: 11
Analytic conductor: 7.57899×10−67.57899\times 10^{-6}
Root analytic conductor: 0.6918110.691811
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 716⋅13716, ( :[0]16), 1)(32,\ 7^{16} \cdot 137^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.28181140340.2818114034
L(12)L(\frac12) ≈\approx 0.28181140340.2818114034
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)
bad7 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}
137 (1+T)16 ( 1 + T )^{16}
good2 (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}
3 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−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}
11 (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}
13 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}
17 (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} )
19 (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} )
23 (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} )
29 (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} )
31 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}
37 (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}
41 (1+T2)16 ( 1 + T^{2} )^{16}
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−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}
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−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} )
61 (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} )
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)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} )
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)(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} )
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−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}
97 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}
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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

−2.83659624188346207892517598100, −2.82067995709621959303502339696, −2.79316767997292805511927804283, −2.69329381089355561226914867858, −2.69092146065958371812964747243, −2.64412711623360736253359120611, −2.43333732129233661685461638866, −2.39181598818617889686586792535, −2.29842816213623331753485952534, −2.24962939761678538199063142631, −2.05488769182451142071173634315, −1.91142803111573887872964263058, −1.73141058467629817108323076608, −1.57083221882918829491113949023, −1.55405214211732689603387849099, −1.53381487883075891578295769059, −1.29554734125878117701305106424, −1.28452899262356262498712570862, −1.25835708961806479113656621189, −1.20292350742541234857455084095, −1.16350869840808422243785762060, −1.09050692018812924224439802498, −1.08480369663458077543599622311, −1.03341601508565436157433883195, −0.65991919945432458549897673333, 0.65991919945432458549897673333, 1.03341601508565436157433883195, 1.08480369663458077543599622311, 1.09050692018812924224439802498, 1.16350869840808422243785762060, 1.20292350742541234857455084095, 1.25835708961806479113656621189, 1.28452899262356262498712570862, 1.29554734125878117701305106424, 1.53381487883075891578295769059, 1.55405214211732689603387849099, 1.57083221882918829491113949023, 1.73141058467629817108323076608, 1.91142803111573887872964263058, 2.05488769182451142071173634315, 2.24962939761678538199063142631, 2.29842816213623331753485952534, 2.39181598818617889686586792535, 2.43333732129233661685461638866, 2.64412711623360736253359120611, 2.69092146065958371812964747243, 2.69329381089355561226914867858, 2.79316767997292805511927804283, 2.82067995709621959303502339696, 2.83659624188346207892517598100

Graph of the ZZ-function along the critical line

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