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Properties

Label 32-884e16-1.1-c0e16-0-1
Degree $32$
Conductor $1.391\times 10^{47}$
Sign $1$
Analytic cond. $2.05944\times 10^{-6}$
Root an. cond. $0.664208$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·29-s − 8·53-s + 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + ⋯
L(s)  = 1  + 8·29-s − 8·53-s + 8·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 13^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(2.05944\times 10^{-6}\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 13^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4638363949\)
\(L(\frac12)\) \(\approx\) \(0.4638363949\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.87930976238846689943392801825, −2.82029779289897477892816208833, −2.75716269425626924879115403088, −2.68942125255138507769682205145, −2.67484247857198001475538168155, −2.50242903189510211828558324598, −2.46685478872041959786092350438, −2.45459317853292150650434710704, −2.44007585398089797582045368665, −2.29204021211967202647872397136, −2.28638156049252468253142693399, −2.10673081777600952742298916992, −1.88225025107850591427041501662, −1.83784293107116822463340349460, −1.82149684593980371713837120766, −1.63117570151215947033392601215, −1.50229519833480092023992526069, −1.47525253806641032483278124680, −1.36547704659359954381515137077, −1.32592017750385745664125234461, −1.05449892289509338374692787430, −1.01720413761775141385137351185, −0.969399870188528542905061957698, −0.955837364243948063021603711216, −0.67658122210769185361920370582, 0.67658122210769185361920370582, 0.955837364243948063021603711216, 0.969399870188528542905061957698, 1.01720413761775141385137351185, 1.05449892289509338374692787430, 1.32592017750385745664125234461, 1.36547704659359954381515137077, 1.47525253806641032483278124680, 1.50229519833480092023992526069, 1.63117570151215947033392601215, 1.82149684593980371713837120766, 1.83784293107116822463340349460, 1.88225025107850591427041501662, 2.10673081777600952742298916992, 2.28638156049252468253142693399, 2.29204021211967202647872397136, 2.44007585398089797582045368665, 2.45459317853292150650434710704, 2.46685478872041959786092350438, 2.50242903189510211828558324598, 2.67484247857198001475538168155, 2.68942125255138507769682205145, 2.75716269425626924879115403088, 2.82029779289897477892816208833, 2.87930976238846689943392801825

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

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