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Properties

Label 16-544e8-1.1-c0e8-0-0
Degree 1616
Conductor 7.670×10217.670\times 10^{21}
Sign 11
Analytic cond. 2.95153×10−52.95153\times 10^{-5}
Root an. cond. 0.5210480.521048
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  − 8·53-s − 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  − 8·53-s − 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯

Functional equation

Λ(s)=((240⋅178)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((240⋅178)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 240⋅1782^{40} \cdot 17^{8}
Sign: 11
Analytic conductor: 2.95153×10−52.95153\times 10^{-5}
Root analytic conductor: 0.5210480.521048
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 240⋅178, ( :[0]8), 1)(16,\ 2^{40} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.30198975940.3019897594
L(12)L(\frac12) ≈\approx 0.30198975940.3019897594
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 1
17 (1+T4)2 ( 1 + T^{4} )^{2}
good3 1+T16 1 + T^{16}
5 (1+T4)2(1+T8) ( 1 + T^{4} )^{2}( 1 + T^{8} )
7 1+T16 1 + T^{16}
11 1+T16 1 + T^{16}
13 (1+T8)2 ( 1 + T^{8} )^{2}
19 (1+T8)2 ( 1 + T^{8} )^{2}
23 1+T16 1 + T^{16}
29 (1+T2)4(1+T8) ( 1 + T^{2} )^{4}( 1 + T^{8} )
31 1+T16 1 + T^{16}
37 (1+T4)2(1+T8) ( 1 + T^{4} )^{2}( 1 + T^{8} )
41 (1+T2)4(1+T8) ( 1 + T^{2} )^{4}( 1 + T^{8} )
43 (1+T8)2 ( 1 + T^{8} )^{2}
47 (1+T4)4 ( 1 + T^{4} )^{4}
53 (1+T)8(1+T4)2 ( 1 + T )^{8}( 1 + T^{4} )^{2}
59 (1+T8)2 ( 1 + T^{8} )^{2}
61 (1+T4)2(1+T8) ( 1 + T^{4} )^{2}( 1 + T^{8} )
67 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
71 1+T16 1 + T^{16}
73 (1+T)8(1+T8) ( 1 + T )^{8}( 1 + T^{8} )
79 1+T16 1 + T^{16}
83 (1+T8)2 ( 1 + T^{8} )^{2}
89 (1+T8)2 ( 1 + T^{8} )^{2}
97 (1+T4)2(1+T8) ( 1 + T^{4} )^{2}( 1 + T^{8} )
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.06239620894146776061362063078, −4.70096464958984022162231711454, −4.57808741067888645370820890259, −4.51072493771622239461044471722, −4.48998568235527809037970386754, −4.35858189279909206451949425978, −4.21763068288587480902041673157, −4.20370402316921004419460849613, −4.10520438691444965839565846077, −3.53372728561153302885629137637, −3.37381165218340105960357164541, −3.34465808366842047067278156145, −3.32720565535012117983580849806, −3.08755474479445968235233227101, −2.94817295402470425033788728748, −2.87912658523284310481614037251, −2.80737861633896188706736572081, −2.53941262515402623269061681640, −2.07373832743468992303578357588, −1.91911946694347466477349107245, −1.75012794395966607867260784287, −1.73852463777248832295259786461, −1.48812981451970089534642670616, −1.36128957131794691255204997358, −0.78114528332852071208586367621, 0.78114528332852071208586367621, 1.36128957131794691255204997358, 1.48812981451970089534642670616, 1.73852463777248832295259786461, 1.75012794395966607867260784287, 1.91911946694347466477349107245, 2.07373832743468992303578357588, 2.53941262515402623269061681640, 2.80737861633896188706736572081, 2.87912658523284310481614037251, 2.94817295402470425033788728748, 3.08755474479445968235233227101, 3.32720565535012117983580849806, 3.34465808366842047067278156145, 3.37381165218340105960357164541, 3.53372728561153302885629137637, 4.10520438691444965839565846077, 4.20370402316921004419460849613, 4.21763068288587480902041673157, 4.35858189279909206451949425978, 4.48998568235527809037970386754, 4.51072493771622239461044471722, 4.57808741067888645370820890259, 4.70096464958984022162231711454, 5.06239620894146776061362063078

Graph of the ZZ-function along the critical line

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