Properties

Degree $32$
Conductor $5.760\times 10^{56}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 6·16-s − 8·25-s − 32·100-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 4·4-s + 6·16-s − 8·25-s − 32·100-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04182009394\)
\(L(\frac12)\) \(\approx\) \(0.04182009394\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{4} \)
3 \( 1 - T^{8} + T^{16} \)
7 \( 1 \)
good5 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
11 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} )^{8} \)
17 \( ( 1 - T^{8} + T^{16} )^{2} \)
19 \( ( 1 - T^{8} + T^{16} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} )^{8} \)
29 \( ( 1 - T^{2} + T^{4} )^{8} \)
31 \( ( 1 - T^{2} + T^{4} )^{8} \)
37 \( ( 1 - T )^{16}( 1 + T )^{16} \)
41 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
43 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
53 \( ( 1 + T^{2} )^{16} \)
59 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
61 \( ( 1 - T^{2} + T^{4} )^{8} \)
67 \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} )^{4} \)
71 \( ( 1 + T^{2} )^{16} \)
73 \( ( 1 - T^{8} + T^{16} )^{2} \)
79 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
83 \( ( 1 - T^{8} + T^{16} )^{2} \)
89 \( ( 1 + T^{8} )^{4} \)
97 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
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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

−2.23488287064216846161298092928, −2.16865519643851876038891156561, −2.08440939855505136947756484737, −2.06540145571525226022308601034, −2.05049025907797447877295683635, −2.00345337489757620449922827930, −2.00121805628386355265720667484, −1.88992426569151772008897104938, −1.86501440951890565720959198219, −1.75835996569407429530150658004, −1.71001700211371802139671784386, −1.70221607101990462068950577605, −1.52775688435436435369141385290, −1.52148995734800611294121360585, −1.40881303361777353639516534741, −1.34336969669496277077143150571, −1.19120073182921632226156055535, −1.13905967885759485009529492018, −0.999743004242491795538438858387, −0.929289765004691417170128259017, −0.921989437552608876905677201784, −0.806895941818325548827357135264, −0.47599565512967195702798966557, −0.38602285949617989978821013844, −0.02644674351811086044867826727, 0.02644674351811086044867826727, 0.38602285949617989978821013844, 0.47599565512967195702798966557, 0.806895941818325548827357135264, 0.921989437552608876905677201784, 0.929289765004691417170128259017, 0.999743004242491795538438858387, 1.13905967885759485009529492018, 1.19120073182921632226156055535, 1.34336969669496277077143150571, 1.40881303361777353639516534741, 1.52148995734800611294121360585, 1.52775688435436435369141385290, 1.70221607101990462068950577605, 1.71001700211371802139671784386, 1.75835996569407429530150658004, 1.86501440951890565720959198219, 1.88992426569151772008897104938, 2.00121805628386355265720667484, 2.00345337489757620449922827930, 2.05049025907797447877295683635, 2.06540145571525226022308601034, 2.08440939855505136947756484737, 2.16865519643851876038891156561, 2.23488287064216846161298092928

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

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