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Properties

Label 32-3267e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.684\times 10^{56}$
Sign $1$
Analytic cond. $2494.02$
Root an. cond. $1.27688$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 6·16-s + 4·25-s − 4·64-s − 4·97-s − 16·100-s − 4·103-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 + ⋯
L(s)  = 1  − 4·4-s + 6·16-s + 4·25-s − 4·64-s − 4·97-s − 16·100-s − 4·103-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{48} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(2494.02\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{48} \cdot 11^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1108564602\)
\(L(\frac12)\) \(\approx\) \(0.1108564602\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
13 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
19 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
23 \( ( 1 + T^{2} )^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 + T^{4} )^{8} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
73 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
79 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
89 \( ( 1 + T^{2} )^{16} \)
97 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.26815971663040867106034850829, −2.23750292159158818923295133690, −2.19168363399318001586107407332, −2.09112482333349634024479048526, −2.08759716635764155310524896208, −2.01720155902631622372864514192, −1.99435865501308734993850452317, −1.89995998755732687185511280733, −1.76802792579404070325287092040, −1.66014619884992120719471127832, −1.65383512614875707759921920346, −1.46741829347561648117891561707, −1.36417890653897213981761366211, −1.25945269566180054280318475513, −1.22554977726817142538065734547, −1.18920143630124494389179214403, −1.17067295121211329376219747123, −1.16619913820001205251355152875, −0.983078397169420318390823652212, −0.924940453784855138517614883580, −0.886188653004514832453301012965, −0.63304588066824351742954666003, −0.43068129681576193715264096459, −0.34669964667033080075886463480, −0.15258689065323235842523057926, 0.15258689065323235842523057926, 0.34669964667033080075886463480, 0.43068129681576193715264096459, 0.63304588066824351742954666003, 0.886188653004514832453301012965, 0.924940453784855138517614883580, 0.983078397169420318390823652212, 1.16619913820001205251355152875, 1.17067295121211329376219747123, 1.18920143630124494389179214403, 1.22554977726817142538065734547, 1.25945269566180054280318475513, 1.36417890653897213981761366211, 1.46741829347561648117891561707, 1.65383512614875707759921920346, 1.66014619884992120719471127832, 1.76802792579404070325287092040, 1.89995998755732687185511280733, 1.99435865501308734993850452317, 2.01720155902631622372864514192, 2.08759716635764155310524896208, 2.09112482333349634024479048526, 2.19168363399318001586107407332, 2.23750292159158818923295133690, 2.26815971663040867106034850829

Graph of the $Z$-function along the critical line

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