Properties

Label 32-943e16-1.1-c0e16-0-0
Degree $32$
Conductor $3.910\times 10^{47}$
Sign $1$
Analytic cond. $5.79030\times 10^{-6}$
Root an. cond. $0.686016$
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  − 2·3-s + 3·4-s + 2·9-s − 6·12-s + 2·13-s + 3·16-s − 4·23-s + 4·25-s − 8·29-s + 6·36-s − 4·39-s + 2·41-s + 2·47-s − 6·48-s + 6·52-s − 8·59-s − 2·64-s + 8·69-s − 2·71-s − 8·75-s − 81-s + 16·87-s − 12·92-s + 12·100-s − 4·101-s − 24·116-s + 4·117-s + ⋯
L(s)  = 1  − 2·3-s + 3·4-s + 2·9-s − 6·12-s + 2·13-s + 3·16-s − 4·23-s + 4·25-s − 8·29-s + 6·36-s − 4·39-s + 2·41-s + 2·47-s − 6·48-s + 6·52-s − 8·59-s − 2·64-s + 8·69-s − 2·71-s − 8·75-s − 81-s + 16·87-s − 12·92-s + 12·100-s − 4·101-s − 24·116-s + 4·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(23^{16} \cdot 41^{16}\)
Sign: $1$
Analytic conductor: \(5.79030\times 10^{-6}\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{943} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 23^{16} \cdot 41^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2407293837\)
\(L(\frac12)\) \(\approx\) \(0.2407293837\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
good2 \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
3 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
7 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
11 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
17 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
19 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 + T + T^{2} )^{8}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
31 \( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{8} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
71 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
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 \( ( 1 + T^{4} )^{8} \)
83 \( ( 1 - T )^{16}( 1 + T )^{16} \)
89 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
97 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
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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

−2.89000792570922135211314014994, −2.82008836860571423314636731758, −2.79332280288089708555501706200, −2.69811017509965093664790945403, −2.59844099779268342328565780217, −2.54830628156530267746049712925, −2.51359092718453813592786809355, −2.27492893653081643326028451790, −2.26550828896855368535095418521, −2.12042993710964606406707275351, −2.00085295430743075985772892434, −1.88887943844739407567196683642, −1.86765192223819035797346627825, −1.84231592711039182908169170396, −1.76753462718111631334388473698, −1.65512285823835893651206760768, −1.62088859145956747480788306046, −1.53451145443997841494593188144, −1.50679053844028745711695509570, −1.33407097231475687058636465056, −1.16689033925713311898573111436, −1.11246441615025548947746899421, −1.00765547252638300073012316760, −0.75855776052480419307337607101, −0.38354315758073013140926621289, 0.38354315758073013140926621289, 0.75855776052480419307337607101, 1.00765547252638300073012316760, 1.11246441615025548947746899421, 1.16689033925713311898573111436, 1.33407097231475687058636465056, 1.50679053844028745711695509570, 1.53451145443997841494593188144, 1.62088859145956747480788306046, 1.65512285823835893651206760768, 1.76753462718111631334388473698, 1.84231592711039182908169170396, 1.86765192223819035797346627825, 1.88887943844739407567196683642, 2.00085295430743075985772892434, 2.12042993710964606406707275351, 2.26550828896855368535095418521, 2.27492893653081643326028451790, 2.51359092718453813592786809355, 2.54830628156530267746049712925, 2.59844099779268342328565780217, 2.69811017509965093664790945403, 2.79332280288089708555501706200, 2.82008836860571423314636731758, 2.89000792570922135211314014994

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

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