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Properties

Label 32-3960e16-1.1-c0e16-0-0
Degree 3232
Conductor 3.657×10573.657\times 10^{57}
Sign 11
Analytic cond. 54154.854154.8
Root an. cond. 1.405801.40580
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  − 4·2-s + 6·4-s − 4·8-s + 16-s + 2·25-s + 4·32-s − 8·50-s − 16·64-s + 12·100-s + 127-s + 24·128-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 − 8·200-s + ⋯
L(s)  = 1  − 4·2-s + 6·4-s − 4·8-s + 16-s + 2·25-s + 4·32-s − 8·50-s − 16·64-s + 12·100-s + 127-s + 24·128-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 − 8·200-s + ⋯

Functional equation

Λ(s)=((248⋅332⋅516⋅1116)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((248⋅332⋅516⋅1116)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 248⋅332⋅516⋅11162^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}
Sign: 11
Analytic conductor: 54154.854154.8
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 248⋅332⋅516⋅1116, ( :[0]16), 1)(32,\ 2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.017547150780.01754715078
L(12)L(\frac12) ≈\approx 0.017547150780.01754715078
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+T+T2+T3+T4)4 ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
3 1 1
5 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
11 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
good7 (1+T4)4(1−T4+T8−T12+T16) ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )
13 (1+T4)4(1−T4+T8−T12+T16) ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )
17 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
19 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
23 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
29 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
31 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
37 (1−T4+T8−T12+T16)2 ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}
41 (1+T4)4(1−T4+T8−T12+T16) ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )
43 (1+T2)16 ( 1 + T^{2} )^{16}
47 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
53 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
59 (1−T4+T8−T12+T16)2 ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}
61 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
67 (1−T)16(1+T)16 ( 1 - T )^{16}( 1 + T )^{16}
71 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
73 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
79 (1−T2+T4−T6+T8)4 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}
83 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}
89 (1−T4+T8−T12+T16)2 ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}
97 (1−T+T2−T3+T4)4(1+T+T2+T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{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

−2.35030793761958260666208541378, −2.26106392409611313642146388499, −2.18134287100420763894814585398, −1.91094678613372930944682346103, −1.87316727869945115540589327442, −1.80750803949519666451552699935, −1.74067404911642514169791343972, −1.71197993070622115989033890852, −1.68603721508571199119059093621, −1.61533556994708509770338364542, −1.49209345724541155130990750778, −1.39260354073603368444943570862, −1.31674983672340694139493010150, −1.27931390574006041131103342001, −1.19461142653265045354560645438, −1.03740206028114100616904381345, −1.02397938545570086339597966092, −1.01855643797560794593288604239, −0.974974480126736226286594487695, −0.969567225421309896446423130931, −0.76441702431114096344915561772, −0.54344277685355246635962499017, −0.38944782631777374307839999932, −0.26608817294915183836396236816, −0.16056266344261937044968692791, 0.16056266344261937044968692791, 0.26608817294915183836396236816, 0.38944782631777374307839999932, 0.54344277685355246635962499017, 0.76441702431114096344915561772, 0.969567225421309896446423130931, 0.974974480126736226286594487695, 1.01855643797560794593288604239, 1.02397938545570086339597966092, 1.03740206028114100616904381345, 1.19461142653265045354560645438, 1.27931390574006041131103342001, 1.31674983672340694139493010150, 1.39260354073603368444943570862, 1.49209345724541155130990750778, 1.61533556994708509770338364542, 1.68603721508571199119059093621, 1.71197993070622115989033890852, 1.74067404911642514169791343972, 1.80750803949519666451552699935, 1.87316727869945115540589327442, 1.91094678613372930944682346103, 2.18134287100420763894814585398, 2.26106392409611313642146388499, 2.35030793761958260666208541378

Graph of the ZZ-function along the critical line

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