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Properties

Label 32-3332e16-1.1-c0e16-0-1
Degree 3232
Conductor 2.308×10562.308\times 10^{56}
Sign 11
Analytic cond. 3418.173418.17
Root an. cond. 1.289521.28952
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  − 16·13-s + 8·37-s + 16·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯
L(s)  = 1  − 16·13-s + 8·37-s + 16·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯

Functional equation

Λ(s)=((232⋅732⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((232⋅732⋅1716)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 232⋅732⋅17162^{32} \cdot 7^{32} \cdot 17^{16}
Sign: 11
Analytic conductor: 3418.173418.17
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 232⋅732⋅1716, ( :[0]16), 1)(32,\ 2^{32} \cdot 7^{32} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58339025420.5833902542
L(12)L(\frac12) ≈\approx 0.58339025420.5833902542
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−T8+T16 1 - T^{8} + T^{16}
7 1 1
17 1−T8+T16 1 - T^{8} + T^{16}
good3 1−T16+T32 1 - T^{16} + T^{32}
5 (1−T4+T8)2(1−T8+T16) ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )
11 1−T16+T32 1 - T^{16} + T^{32}
13 (1+T)16(1+T2)8 ( 1 + T )^{16}( 1 + T^{2} )^{8}
19 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
23 1−T16+T32 1 - T^{16} + T^{32}
29 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}
31 1−T16+T32 1 - T^{16} + T^{32}
37 (1−T+T2)8(1−T8+T16) ( 1 - T + T^{2} )^{8}( 1 - T^{8} + T^{16} )
41 (1−T)16(1+T8)2 ( 1 - T )^{16}( 1 + T^{8} )^{2}
43 (1+T8)4 ( 1 + T^{8} )^{4}
47 (1−T4+T8)4 ( 1 - T^{4} + T^{8} )^{4}
53 (1−T2+T4)4(1−T4+T8)2 ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2}
59 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
61 (1−T+T2)8(1−T8+T16) ( 1 - T + T^{2} )^{8}( 1 - T^{8} + T^{16} )
67 (1−T2+T4)8 ( 1 - T^{2} + T^{4} )^{8}
71 (1+T16)2 ( 1 + T^{16} )^{2}
73 (1−T4+T8)2(1−T8+T16) ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )
79 1−T16+T32 1 - T^{16} + T^{32}
83 (1+T8)4 ( 1 + T^{8} )^{4}
89 (1−T4+T8)4 ( 1 - T^{4} + T^{8} )^{4}
97 (1+T4)4(1+T8)2 ( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}
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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

−2.46949338274332776133257898387, −2.44690739075334442720094291530, −2.40032011223918237020799365135, −2.29394091253661954429630664420, −2.19459518481256367512294349062, −2.10837191417237831268563741099, −2.06395770512259787839920617769, −2.01713412011958660145452404471, −1.78608899108506902513280801517, −1.77000008332226694898470075797, −1.75401684186257254299049495232, −1.68702568768526371249868466462, −1.66534477869315113410535826088, −1.24826894549178629095598830741, −1.22464392656625361570815943220, −1.15541141622206115605994105893, −1.06352760584071827954714501542, −1.01198060239078931750983454278, −0.977053880368121569984014656510, −0.819842699255900549429978681864, −0.64990024209747212235750578380, −0.60367659484654544932842997203, −0.59823070558453120155209330107, −0.48229656448719361807815991016, −0.22147116485545379211991940325, 0.22147116485545379211991940325, 0.48229656448719361807815991016, 0.59823070558453120155209330107, 0.60367659484654544932842997203, 0.64990024209747212235750578380, 0.819842699255900549429978681864, 0.977053880368121569984014656510, 1.01198060239078931750983454278, 1.06352760584071827954714501542, 1.15541141622206115605994105893, 1.22464392656625361570815943220, 1.24826894549178629095598830741, 1.66534477869315113410535826088, 1.68702568768526371249868466462, 1.75401684186257254299049495232, 1.77000008332226694898470075797, 1.78608899108506902513280801517, 2.01713412011958660145452404471, 2.06395770512259787839920617769, 2.10837191417237831268563741099, 2.19459518481256367512294349062, 2.29394091253661954429630664420, 2.40032011223918237020799365135, 2.44690739075334442720094291530, 2.46949338274332776133257898387

Graph of the ZZ-function along the critical line

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