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Properties

Label 32-3528e16-1.1-c0e16-0-1
Degree $32$
Conductor $5.760\times 10^{56}$
Sign $1$
Analytic cond. $8530.43$
Root an. cond. $1.32691$
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  − 8·4-s + 36·16-s − 8·25-s − 120·64-s + 64·100-s − 24·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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  − 8·4-s + 36·16-s − 8·25-s − 120·64-s + 64·100-s − 24·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(8530.43\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004491218966\)
\(L(\frac12)\) \(\approx\) \(0.004491218966\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{8} \)
3 \( 1 - T^{8} + T^{16} \)
7 \( 1 \)
good5 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
11 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} )^{8} \)
17 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
19 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
23 \( ( 1 - T^{2} + T^{4} )^{8} \)
29 \( ( 1 - T^{2} + T^{4} )^{8} \)
31 \( ( 1 + T^{2} )^{16} \)
37 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
41 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
43 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T )^{16}( 1 + T )^{16} \)
53 \( ( 1 - T^{2} + T^{4} )^{8} \)
59 \( ( 1 - T^{8} + T^{16} )^{2} \)
61 \( ( 1 + T^{2} )^{16} \)
67 \( ( 1 - T^{2} + T^{4} )^{8} \)
71 \( ( 1 + T^{2} )^{16} \)
73 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
79 \( ( 1 - T )^{16}( 1 + T )^{16} \)
83 \( ( 1 - T^{8} + T^{16} )^{2} \)
89 \( ( 1 - T^{8} + T^{16} )^{2} \)
97 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
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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

−2.25014940926321746274229140812, −2.23153992826095682723948355867, −2.12153288109858573081293592223, −1.96235592365060485043137817571, −1.94504590333269035776144020908, −1.85605511824035031292464250997, −1.80631224634585544729146335804, −1.77092645283471072094192684338, −1.68608673502969915006567851202, −1.60178836186570864769219561240, −1.58934303214049832818476805072, −1.46532298241489947475730835426, −1.42044875001478911962576463015, −1.34193865002238550815484174361, −1.33000871933692912725118811023, −1.18044967447081141471602298490, −1.04145783288098238768368388651, −1.03525561396293075791187793925, −1.01297716075002407220536131696, −0.75866091524924606614672323239, −0.67472648994210676572252562389, −0.58734762036789684022756430096, −0.33925969743452682726286605767, −0.19504993542055297238883079821, −0.10446153697713211463722266382, 0.10446153697713211463722266382, 0.19504993542055297238883079821, 0.33925969743452682726286605767, 0.58734762036789684022756430096, 0.67472648994210676572252562389, 0.75866091524924606614672323239, 1.01297716075002407220536131696, 1.03525561396293075791187793925, 1.04145783288098238768368388651, 1.18044967447081141471602298490, 1.33000871933692912725118811023, 1.34193865002238550815484174361, 1.42044875001478911962576463015, 1.46532298241489947475730835426, 1.58934303214049832818476805072, 1.60178836186570864769219561240, 1.68608673502969915006567851202, 1.77092645283471072094192684338, 1.80631224634585544729146335804, 1.85605511824035031292464250997, 1.94504590333269035776144020908, 1.96235592365060485043137817571, 2.12153288109858573081293592223, 2.23153992826095682723948355867, 2.25014940926321746274229140812

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

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