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Properties

Label 64-3525e32-1.1-c0e32-0-0
Degree $64$
Conductor $3.229\times 10^{113}$
Sign $1$
Analytic cond. $7.08141\times 10^{7}$
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 + 16·61-s − 81-s − 32·121-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 + 16·61-s − 81-s − 32·121-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{64} \cdot 47^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{64} \cdot 47^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(64\)
Conductor: \(3^{32} \cdot 5^{64} \cdot 47^{32}\)
Sign: $1$
Analytic conductor: \(7.08141\times 10^{7}\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((64,\ 3^{32} \cdot 5^{64} \cdot 47^{32} ,\ ( \ : [0]^{32} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5185376064\)
\(L(\frac12)\) \(\approx\) \(0.5185376064\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
5 \( 1 \)
47 \( ( 1 + T^{4} )^{8} \)
good2 \( ( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} )^{2} \)
7 \( ( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} )^{2} \)
11 \( ( 1 + T^{2} )^{32} \)
13 \( ( 1 + T^{4} )^{16} \)
17 \( ( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} )^{2} \)
19 \( ( 1 + T^{2} )^{32} \)
23 \( ( 1 + T^{4} )^{16} \)
29 \( ( 1 - T )^{32}( 1 + T )^{32} \)
31 \( ( 1 - T )^{32}( 1 + T )^{32} \)
37 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{4} \)
41 \( ( 1 + T^{2} )^{32} \)
43 \( ( 1 + T^{4} )^{16} \)
53 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{4} \)
59 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{4} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{16} \)
67 \( ( 1 + T^{4} )^{16} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{4} \)
73 \( ( 1 + T^{4} )^{16} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{8} \)
83 \( ( 1 - T^{4} + T^{8} )^{8} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{8} \)
97 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.43849918948851363304652964546, −1.42345122762625830126732431736, −1.38682880217682400394396460237, −1.27011754978361545487989563046, −1.23388347317508550450344671230, −1.23294626106974856791672608254, −1.18777318081906914367997130618, −1.17730178728254094311885427865, −1.16338293486493229686563358410, −1.11706554694315763733255365883, −1.06229397654258010213469038231, −1.03904083144915631350885993247, −1.03186691184152933931571805280, −1.01204117989195723452752370805, −0.950402859859755222489448082116, −0.797523592106154471875387538245, −0.77027496400294509164710637853, −0.73052887900039050933833405381, −0.69331447845506268889089306148, −0.61915287531945256078156126575, −0.61387806307452345091688026962, −0.38152149203968294188715136713, −0.32686234025722817679858338553, −0.23821286751120918518821922467, −0.11496456367563553406290749908, 0.11496456367563553406290749908, 0.23821286751120918518821922467, 0.32686234025722817679858338553, 0.38152149203968294188715136713, 0.61387806307452345091688026962, 0.61915287531945256078156126575, 0.69331447845506268889089306148, 0.73052887900039050933833405381, 0.77027496400294509164710637853, 0.797523592106154471875387538245, 0.950402859859755222489448082116, 1.01204117989195723452752370805, 1.03186691184152933931571805280, 1.03904083144915631350885993247, 1.06229397654258010213469038231, 1.11706554694315763733255365883, 1.16338293486493229686563358410, 1.17730178728254094311885427865, 1.18777318081906914367997130618, 1.23294626106974856791672608254, 1.23388347317508550450344671230, 1.27011754978361545487989563046, 1.38682880217682400394396460237, 1.42345122762625830126732431736, 1.43849918948851363304652964546

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

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