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Properties

Label 32-579e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.595\times 10^{44}$
Sign $1$
Analytic cond. $2.36249\times 10^{-9}$
Root an. cond. $0.537548$
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 + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-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 + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  − 16·13-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-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 + 241-s + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 193^{16}\)
Sign: $1$
Analytic conductor: \(2.36249\times 10^{-9}\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 193^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01305291948\)
\(L(\frac12)\) \(\approx\) \(0.01305291948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{8} )^{2} \)
193 \( ( 1 + T^{8} )^{2} \)
good2 \( ( 1 + T^{16} )^{2} \)
5 \( 1 + T^{32} \)
7 \( ( 1 + T^{16} )^{2} \)
11 \( 1 + T^{32} \)
13 \( ( 1 + T )^{16}( 1 + T^{16} ) \)
17 \( 1 + T^{32} \)
19 \( ( 1 + T^{4} )^{4}( 1 + T^{16} ) \)
23 \( ( 1 + T^{16} )^{2} \)
29 \( 1 + T^{32} \)
31 \( ( 1 + T^{16} )^{2} \)
37 \( ( 1 + T^{2} )^{8}( 1 + T^{16} ) \)
41 \( 1 + T^{32} \)
43 \( ( 1 + T^{16} )^{2} \)
47 \( 1 + T^{32} \)
53 \( 1 + T^{32} \)
59 \( ( 1 + T^{8} )^{4} \)
61 \( ( 1 + T^{2} )^{8}( 1 + T^{16} ) \)
67 \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \)
71 \( 1 + T^{32} \)
73 \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \)
79 \( ( 1 + T^{8} )^{2}( 1 + T^{16} ) \)
83 \( ( 1 + T^{16} )^{2} \)
89 \( 1 + T^{32} \)
97 \( ( 1 + T^{16} )^{2} \)
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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

−3.14008141214671876647346594448, −3.06965488896044187763369207869, −2.98152666404592363843685980755, −2.83748525198955354274915720162, −2.74496134823515485424242581257, −2.66765771924745179308051197213, −2.61428054061799489162970826165, −2.55570480063021772309209560958, −2.47465184985798324585479870430, −2.38204597958068103019312494731, −2.36062547143334541703062016245, −2.34148875325311238792463247072, −2.29359240125934235870376555445, −2.23757215600066676640908265058, −2.21027825867868094838694125680, −2.03241737971293993293486947485, −2.00612519443209139350884772468, −1.65959156661330966386733070173, −1.56142756919976868188868371924, −1.47272848226435103116555097572, −1.47213214605567126917006251392, −1.44716660260406667626755650158, −0.895697912928697223669291142032, −0.65141836869973451084679036472, −0.29834055462519845325937510912, 0.29834055462519845325937510912, 0.65141836869973451084679036472, 0.895697912928697223669291142032, 1.44716660260406667626755650158, 1.47213214605567126917006251392, 1.47272848226435103116555097572, 1.56142756919976868188868371924, 1.65959156661330966386733070173, 2.00612519443209139350884772468, 2.03241737971293993293486947485, 2.21027825867868094838694125680, 2.23757215600066676640908265058, 2.29359240125934235870376555445, 2.34148875325311238792463247072, 2.36062547143334541703062016245, 2.38204597958068103019312494731, 2.47465184985798324585479870430, 2.55570480063021772309209560958, 2.61428054061799489162970826165, 2.66765771924745179308051197213, 2.74496134823515485424242581257, 2.83748525198955354274915720162, 2.98152666404592363843685980755, 3.06965488896044187763369207869, 3.14008141214671876647346594448

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

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