Properties

Label 32-3332e16-1.1-c0e16-0-1
Degree $32$
Conductor $2.308\times 10^{56}$
Sign $1$
Analytic cond. $3418.17$
Root an. cond. $1.28952$
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  − 16·13-s + 8·37-s + 16·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯
L(s)  = 1  − 16·13-s + 8·37-s + 16·41-s + 8·61-s − 16·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{32} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 7^{32} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(3418.17\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 7^{32} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5833902542\)
\(L(\frac12)\) \(\approx\) \(0.5833902542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.46949338274332776133257898387, −2.44690739075334442720094291530, −2.40032011223918237020799365135, −2.29394091253661954429630664420, −2.19459518481256367512294349062, −2.10837191417237831268563741099, −2.06395770512259787839920617769, −2.01713412011958660145452404471, −1.78608899108506902513280801517, −1.77000008332226694898470075797, −1.75401684186257254299049495232, −1.68702568768526371249868466462, −1.66534477869315113410535826088, −1.24826894549178629095598830741, −1.22464392656625361570815943220, −1.15541141622206115605994105893, −1.06352760584071827954714501542, −1.01198060239078931750983454278, −0.977053880368121569984014656510, −0.819842699255900549429978681864, −0.64990024209747212235750578380, −0.60367659484654544932842997203, −0.59823070558453120155209330107, −0.48229656448719361807815991016, −0.22147116485545379211991940325, 0.22147116485545379211991940325, 0.48229656448719361807815991016, 0.59823070558453120155209330107, 0.60367659484654544932842997203, 0.64990024209747212235750578380, 0.819842699255900549429978681864, 0.977053880368121569984014656510, 1.01198060239078931750983454278, 1.06352760584071827954714501542, 1.15541141622206115605994105893, 1.22464392656625361570815943220, 1.24826894549178629095598830741, 1.66534477869315113410535826088, 1.68702568768526371249868466462, 1.75401684186257254299049495232, 1.77000008332226694898470075797, 1.78608899108506902513280801517, 2.01713412011958660145452404471, 2.06395770512259787839920617769, 2.10837191417237831268563741099, 2.19459518481256367512294349062, 2.29394091253661954429630664420, 2.40032011223918237020799365135, 2.44690739075334442720094291530, 2.46949338274332776133257898387

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

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