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Properties

Label 32-3525e16-1.1-c0e16-0-0
Degree $32$
Conductor $5.683\times 10^{56}$
Sign $1$
Analytic cond. $8415.11$
Root an. cond. $1.32634$
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  + 2·16-s + 8·61-s + 81-s − 16·121-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 + ⋯
L(s)  = 1  + 2·16-s + 8·61-s + 81-s − 16·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 47^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 47^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 5^{32} \cdot 47^{16}\)
Sign: $1$
Analytic conductor: \(8415.11\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 5^{32} \cdot 47^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.537122216\)
\(L(\frac12)\) \(\approx\) \(3.537122216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.29893919704095474985767140018, −2.23697246251203749123268717835, −2.19931726441134502306091423299, −2.16112232666304044760086889760, −2.10168817394705730961119653307, −2.01092967131162696508748563037, −1.96544144524055926533076938016, −1.78079392583461774626607168626, −1.75539062138956730014176721173, −1.61947575997453879947000540761, −1.55832285615854865780317601093, −1.54802855020588051035078206144, −1.48650650610145134318395422319, −1.36216423164237213625758795417, −1.31213442405187358351663367009, −1.27530065763881183560254966616, −1.11550402676002248517767201925, −1.11016132844291989238762394565, −1.05980183717355881867326324300, −0.859706433913556267121193979715, −0.849682442915890559267681909081, −0.68397582744132200277971786677, −0.54659724276299136973108284035, −0.54417000979890595342875622833, −0.28829615055330846065024616121, 0.28829615055330846065024616121, 0.54417000979890595342875622833, 0.54659724276299136973108284035, 0.68397582744132200277971786677, 0.849682442915890559267681909081, 0.859706433913556267121193979715, 1.05980183717355881867326324300, 1.11016132844291989238762394565, 1.11550402676002248517767201925, 1.27530065763881183560254966616, 1.31213442405187358351663367009, 1.36216423164237213625758795417, 1.48650650610145134318395422319, 1.54802855020588051035078206144, 1.55832285615854865780317601093, 1.61947575997453879947000540761, 1.75539062138956730014176721173, 1.78079392583461774626607168626, 1.96544144524055926533076938016, 2.01092967131162696508748563037, 2.10168817394705730961119653307, 2.16112232666304044760086889760, 2.19931726441134502306091423299, 2.23697246251203749123268717835, 2.29893919704095474985767140018

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

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