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Properties

Label 32-984e16-1.1-c0e16-0-1
Degree $32$
Conductor $7.725\times 10^{47}$
Sign $1$
Analytic cond. $1.14402\times 10^{-5}$
Root an. cond. $0.700770$
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  − 4·3-s + 6·9-s + 16-s + 4·17-s − 4·27-s − 4·48-s − 16·51-s − 16·67-s + 81-s + 4·89-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4·3-s + 6·9-s + 16-s + 4·17-s − 4·27-s − 4·48-s − 16·51-s − 16·67-s + 81-s + 4·89-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 41^{16}\)
Sign: $1$
Analytic conductor: \(1.14402\times 10^{-5}\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{16} \cdot 41^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06213162691\)
\(L(\frac12)\) \(\approx\) \(0.06213162691\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
41 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good5 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
7 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
11 \( ( 1 + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
19 \( ( 1 + T^{2} )^{8}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
29 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
47 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
53 \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
67 \( ( 1 + T )^{16}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
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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

−2.97578895965750578306977020250, −2.94735650782143123331770181642, −2.69135851373024059624198269353, −2.66969969927961625032065184865, −2.61882927134268631539943010587, −2.59921426267285914359632876367, −2.50661051624844243191407308493, −2.30663092180870447473393523714, −2.27301522081470766190512000892, −2.12336391843154232107267260688, −1.92673301335269606055719707329, −1.86349957019538516759405936273, −1.82866656126462789938384513104, −1.68598565428705395219361286779, −1.58545044524254844062421067484, −1.50414767491397077673289440772, −1.46766719794952320197551160663, −1.45555405333513696643985266645, −1.43552165517408539887573819544, −1.40547329776343704825756767938, −0.873926974778355221834369820291, −0.851669057308698973672044917377, −0.849300065948578793238487911894, −0.73450578042483804034159042729, −0.41649767305407453135186592383, 0.41649767305407453135186592383, 0.73450578042483804034159042729, 0.849300065948578793238487911894, 0.851669057308698973672044917377, 0.873926974778355221834369820291, 1.40547329776343704825756767938, 1.43552165517408539887573819544, 1.45555405333513696643985266645, 1.46766719794952320197551160663, 1.50414767491397077673289440772, 1.58545044524254844062421067484, 1.68598565428705395219361286779, 1.82866656126462789938384513104, 1.86349957019538516759405936273, 1.92673301335269606055719707329, 2.12336391843154232107267260688, 2.27301522081470766190512000892, 2.30663092180870447473393523714, 2.50661051624844243191407308493, 2.59921426267285914359632876367, 2.61882927134268631539943010587, 2.66969969927961625032065184865, 2.69135851373024059624198269353, 2.94735650782143123331770181642, 2.97578895965750578306977020250

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

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