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Properties

Label 32-3960e16-1.1-c0e16-0-0
Degree $32$
Conductor $3.657\times 10^{57}$
Sign $1$
Analytic cond. $54154.8$
Root an. cond. $1.40580$
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  − 4·2-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(54154.8\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{32} \cdot 5^{16} \cdot 11^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01754715078\)
\(L(\frac12)\) \(\approx\) \(0.01754715078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
3 \( 1 \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good7 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
13 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
41 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
43 \( ( 1 + T^{2} )^{16} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 - T )^{16}( 1 + T )^{16} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
89 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
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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

−2.35030793761958260666208541378, −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 $Z$-function along the critical line

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