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Properties

Label 32-3660e16-1.1-c0e16-0-2
Degree $32$
Conductor $1.037\times 10^{57}$
Sign $1$
Analytic cond. $15353.6$
Root an. cond. $1.35150$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 3·4-s − 8·6-s + 2·8-s + 6·9-s − 12·12-s + 16-s − 2·17-s + 12·18-s − 2·23-s − 8·24-s − 25-s − 4·27-s + 4·31-s − 2·32-s − 4·34-s + 18·36-s − 4·46-s − 4·48-s − 2·50-s + 8·51-s − 2·53-s − 8·54-s + 8·62-s − 4·64-s − 6·68-s + ⋯
L(s)  = 1  + 2·2-s − 4·3-s + 3·4-s − 8·6-s + 2·8-s + 6·9-s − 12·12-s + 16-s − 2·17-s + 12·18-s − 2·23-s − 8·24-s − 25-s − 4·27-s + 4·31-s − 2·32-s − 4·34-s + 18·36-s − 4·46-s − 4·48-s − 2·50-s + 8·51-s − 2·53-s − 8·54-s + 8·62-s − 4·64-s − 6·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 61^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 61^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 61^{16}\)
Sign: $1$
Analytic conductor: \(15353.6\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 61^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1071800530\)
\(L(\frac12)\) \(\approx\) \(0.1071800530\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
5 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
61 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
good7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
11 \( ( 1 + T^{4} )^{8} \)
13 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
29 \( ( 1 - T^{4} + T^{8} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
37 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
43 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
47 \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} )^{4} \)
53 \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
59 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
67 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
71 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
89 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
97 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
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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

−2.31912183382019807291176032816, −2.20965692165688808024531905239, −2.20162664570455923941875883691, −2.09106636964209215993078597229, −2.05791920381512762982862944636, −1.99834560480301877584479882972, −1.99732315808733553506832139194, −1.93930216876560395610471382687, −1.70085917273180922740426649290, −1.67878381682384829431897057471, −1.55110046941482918892866706883, −1.53175855252283905864031375189, −1.47373355474868672473793434056, −1.42313050419226103303460194104, −1.38517150023827409676395009864, −1.15573016876962775181542341027, −1.12518952360552610780291935481, −1.00939866272458158434239695742, −0.911538765026811372145638989814, −0.864242299756970542293294481547, −0.789902144350179311759656630261, −0.65504429954693804005477449574, −0.41253064034645557540962005771, −0.19421083012914201770195092386, −0.17788465903478849403636933573, 0.17788465903478849403636933573, 0.19421083012914201770195092386, 0.41253064034645557540962005771, 0.65504429954693804005477449574, 0.789902144350179311759656630261, 0.864242299756970542293294481547, 0.911538765026811372145638989814, 1.00939866272458158434239695742, 1.12518952360552610780291935481, 1.15573016876962775181542341027, 1.38517150023827409676395009864, 1.42313050419226103303460194104, 1.47373355474868672473793434056, 1.53175855252283905864031375189, 1.55110046941482918892866706883, 1.67878381682384829431897057471, 1.70085917273180922740426649290, 1.93930216876560395610471382687, 1.99732315808733553506832139194, 1.99834560480301877584479882972, 2.05791920381512762982862944636, 2.09106636964209215993078597229, 2.20162664570455923941875883691, 2.20965692165688808024531905239, 2.31912183382019807291176032816

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

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