Properties

Label 32-975e16-1.1-c0e16-0-1
Degree $32$
Conductor $6.669\times 10^{47}$
Sign $1$
Analytic cond. $9.87615\times 10^{-6}$
Root an. cond. $0.697558$
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  − 2·4-s − 8·7-s − 9-s + 4·13-s + 3·16-s + 2·19-s + 2·25-s + 16·28-s + 2·36-s − 2·37-s + 4·43-s + 36·49-s − 8·52-s + 2·61-s + 8·63-s − 2·64-s + 4·67-s + 4·73-s − 4·76-s + 81-s − 32·91-s − 4·97-s − 4·100-s + 12·103-s − 24·112-s − 4·117-s − 2·121-s + ⋯
L(s)  = 1  − 2·4-s − 8·7-s − 9-s + 4·13-s + 3·16-s + 2·19-s + 2·25-s + 16·28-s + 2·36-s − 2·37-s + 4·43-s + 36·49-s − 8·52-s + 2·61-s + 8·63-s − 2·64-s + 4·67-s + 4·73-s − 4·76-s + 81-s − 32·91-s − 4·97-s − 4·100-s + 12·103-s − 24·112-s − 4·117-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 5^{32} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(9.87615\times 10^{-6}\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{975} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 5^{32} \cdot 13^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1204497649\)
\(L(\frac12)\) \(\approx\) \(0.1204497649\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.10819767468947019993254109937, −2.77275081455260490849373842914, −2.62789892111426265458260752229, −2.56390712520780904835665747435, −2.54383275061172417970706335580, −2.53951155992785152572156175193, −2.51370663415033563705692659390, −2.49866972560658197034910850023, −2.46350405951624965530189600947, −2.34123269494058242456568465982, −2.28420086772296671246306701482, −2.06338271119473914449267392653, −2.03497048869501770690952500278, −1.94892593838254745935304213732, −1.78936994083780491174032500563, −1.36226444086336558406714370929, −1.34833703054177460407003292661, −1.29358138860036969335712234091, −1.25855475839182839031994632710, −1.13196662942483660386211885773, −0.925519101528312968271274583514, −0.874087784466937982936765796443, −0.77279085316847595321165205503, −0.76847739084472296387364214975, −0.59673232488924915026610090919, 0.59673232488924915026610090919, 0.76847739084472296387364214975, 0.77279085316847595321165205503, 0.874087784466937982936765796443, 0.925519101528312968271274583514, 1.13196662942483660386211885773, 1.25855475839182839031994632710, 1.29358138860036969335712234091, 1.34833703054177460407003292661, 1.36226444086336558406714370929, 1.78936994083780491174032500563, 1.94892593838254745935304213732, 2.03497048869501770690952500278, 2.06338271119473914449267392653, 2.28420086772296671246306701482, 2.34123269494058242456568465982, 2.46350405951624965530189600947, 2.49866972560658197034910850023, 2.51370663415033563705692659390, 2.53951155992785152572156175193, 2.54383275061172417970706335580, 2.56390712520780904835665747435, 2.62789892111426265458260752229, 2.77275081455260490849373842914, 3.10819767468947019993254109937

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

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