Properties

Label 32-723e16-1.1-c0e16-0-0
Degree $32$
Conductor $5.575\times 10^{45}$
Sign $1$
Analytic cond. $8.25518\times 10^{-8}$
Root an. cond. $0.600686$
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  − 8·16-s − 4·31-s + 4·43-s + 2·49-s − 4·73-s + 81-s − 4·109-s + 127-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 8·16-s − 4·31-s + 4·43-s + 2·49-s − 4·73-s + 81-s − 4·109-s + 127-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 241^{16}\)
Sign: $1$
Analytic conductor: \(8.25518\times 10^{-8}\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{723} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 241^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09625169364\)
\(L(\frac12)\) \(\approx\) \(0.09625169364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
241 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good2 \( ( 1 + T^{4} )^{8} \)
5 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
11 \( ( 1 + T^{8} )^{4} \)
13 \( ( 1 + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
23 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
29 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
41 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
67 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
71 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
79 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
89 \( ( 1 + T^{8} )^{4} \)
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

−2.95058599179678617891973105398, −2.93185861444184644926174672958, −2.81656968986565822336893575832, −2.79265021463431918563890455634, −2.76734462571796161873970638185, −2.73721776144824284606854038178, −2.65882660046340465268640877301, −2.46536572948297363872433937051, −2.26700261175387134565059522331, −2.22828770461585235388507943151, −2.18529203568086938100292936089, −2.15812642129919159558977290088, −2.05976409916572979869435350999, −1.99465812269000676796825668164, −1.95267202580900828802467861774, −1.93994628129376184274284164000, −1.78369442696156711910489067142, −1.68119734513961924150749710037, −1.58839849175731181443384413306, −1.39667898996689312948828851651, −1.25223685278196553726734775635, −1.02622309788362168630356277211, −1.01271942470610658630927416191, −0.78153479464684803768118330561, −0.44872896704561440138061203752, 0.44872896704561440138061203752, 0.78153479464684803768118330561, 1.01271942470610658630927416191, 1.02622309788362168630356277211, 1.25223685278196553726734775635, 1.39667898996689312948828851651, 1.58839849175731181443384413306, 1.68119734513961924150749710037, 1.78369442696156711910489067142, 1.93994628129376184274284164000, 1.95267202580900828802467861774, 1.99465812269000676796825668164, 2.05976409916572979869435350999, 2.15812642129919159558977290088, 2.18529203568086938100292936089, 2.22828770461585235388507943151, 2.26700261175387134565059522331, 2.46536572948297363872433937051, 2.65882660046340465268640877301, 2.73721776144824284606854038178, 2.76734462571796161873970638185, 2.79265021463431918563890455634, 2.81656968986565822336893575832, 2.93185861444184644926174672958, 2.95058599179678617891973105398

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

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