Properties

Label 32-820e16-1.1-c0e16-0-1
Degree $32$
Conductor $4.179\times 10^{46}$
Sign $1$
Analytic cond. $6.18779\times 10^{-7}$
Root an. cond. $0.639713$
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  + 4·2-s + 6·4-s + 4·8-s + 16-s + 4·29-s − 4·32-s + 16·58-s − 4·61-s − 16·64-s + 4·97-s + 4·109-s + 24·116-s − 16·122-s + 127-s − 24·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4·2-s + 6·4-s + 4·8-s + 16-s + 4·29-s − 4·32-s + 16·58-s − 4·61-s − 16·64-s + 4·97-s + 4·109-s + 24·116-s − 16·122-s + 127-s − 24·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 5^{16} \cdot 41^{16}\)
Sign: $1$
Analytic conductor: \(6.18779\times 10^{-7}\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{820} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 5^{16} \cdot 41^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.870350851\)
\(L(\frac12)\) \(\approx\) \(1.870350851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
good3 \( ( 1 + T^{8} )^{4} \)
7 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
11 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
19 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
23 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
37 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
47 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
71 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( ( 1 + T^{4} )^{8} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T^{4} )^{4} \)
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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

−3.16277622118364898217085602746, −2.98081962808169332441617228833, −2.88799173453384938105111012110, −2.71995820173523354269053302274, −2.64949707874236024026379620298, −2.58629967143707761821798279230, −2.57124118521226436510873457139, −2.53456020977500189266346980896, −2.47104627426295601487773211216, −2.37703782299849093310906873650, −2.29665218294532170545699391445, −2.18851877275340533226804231761, −1.99832976271846608896041872361, −1.95211549590788758835574713936, −1.91606317649761091489691175426, −1.88860688591616982141893676877, −1.63097389078000598161265463394, −1.52337743345770967250874071869, −1.51390013658146603808511292252, −1.27181600472991252701223848036, −1.23014123901706028452141034605, −1.17273397635907200023516797861, −1.07626298196868438396794698668, −0.935923241922551341564064401155, −0.66407545879338416225405512040, 0.66407545879338416225405512040, 0.935923241922551341564064401155, 1.07626298196868438396794698668, 1.17273397635907200023516797861, 1.23014123901706028452141034605, 1.27181600472991252701223848036, 1.51390013658146603808511292252, 1.52337743345770967250874071869, 1.63097389078000598161265463394, 1.88860688591616982141893676877, 1.91606317649761091489691175426, 1.95211549590788758835574713936, 1.99832976271846608896041872361, 2.18851877275340533226804231761, 2.29665218294532170545699391445, 2.37703782299849093310906873650, 2.47104627426295601487773211216, 2.53456020977500189266346980896, 2.57124118521226436510873457139, 2.58629967143707761821798279230, 2.64949707874236024026379620298, 2.71995820173523354269053302274, 2.88799173453384938105111012110, 2.98081962808169332441617228833, 3.16277622118364898217085602746

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

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