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Properties

Label 32-2013e16-1.1-c0e16-0-0
Degree $32$
Conductor $7.269\times 10^{52}$
Sign $1$
Analytic cond. $1.07650$
Root an. cond. $1.00230$
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  + 4·3-s + 6·9-s − 3·16-s − 4·25-s + 4·27-s − 12·48-s − 4·49-s + 4·61-s + 8·73-s − 16·75-s + 81-s + 8·109-s + 127-s + 131-s + 137-s + 139-s − 18·144-s − 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4·3-s + 6·9-s − 3·16-s − 4·25-s + 4·27-s − 12·48-s − 4·49-s + 4·61-s + 8·73-s − 16·75-s + 81-s + 8·109-s + 127-s + 131-s + 137-s + 139-s − 18·144-s − 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16} \cdot 61^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16} \cdot 61^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 11^{16} \cdot 61^{16}\)
Sign: $1$
Analytic conductor: \(1.07650\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 11^{16} \cdot 61^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.453383409\)
\(L(\frac12)\) \(\approx\) \(3.453383409\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.45077683731948893226885205996, −2.41462667961955188958378324491, −2.40853350610774955147545431173, −2.30387307424830489558158830273, −2.28857873144811104849140819493, −2.28072648475148568048610636940, −2.27177855063693084980982456094, −2.17782649488023490259241029888, −1.95066699871069439480579623908, −1.90944607994317792711097544702, −1.87347397251295580593681534159, −1.83560640296761417309634171290, −1.79171537658583323397369183110, −1.74917875758040828638643405244, −1.73252143870362796964371001492, −1.38384935506887729006728618218, −1.36990127099163908495953238378, −1.28312408580997660719761067881, −1.20729025660228412175804394781, −1.06291755426507136053271374538, −0.929316804963404035556181063398, −0.908207317474104170195334532246, −0.62717365881658090889106592400, −0.55111017882565475158532499638, −0.36115825192076849760638840886, 0.36115825192076849760638840886, 0.55111017882565475158532499638, 0.62717365881658090889106592400, 0.908207317474104170195334532246, 0.929316804963404035556181063398, 1.06291755426507136053271374538, 1.20729025660228412175804394781, 1.28312408580997660719761067881, 1.36990127099163908495953238378, 1.38384935506887729006728618218, 1.73252143870362796964371001492, 1.74917875758040828638643405244, 1.79171537658583323397369183110, 1.83560640296761417309634171290, 1.87347397251295580593681534159, 1.90944607994317792711097544702, 1.95066699871069439480579623908, 2.17782649488023490259241029888, 2.27177855063693084980982456094, 2.28072648475148568048610636940, 2.28857873144811104849140819493, 2.30387307424830489558158830273, 2.40853350610774955147545431173, 2.41462667961955188958378324491, 2.45077683731948893226885205996

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

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