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Properties

Label 32-964e16-1.1-c0e16-0-0
Degree $32$
Conductor $5.562\times 10^{47}$
Sign $1$
Analytic cond. $8.23657\times 10^{-6}$
Root an. cond. $0.693612$
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  + 4·4-s − 2·9-s + 4·13-s + 6·16-s − 8·17-s − 2·25-s − 8·36-s + 2·37-s + 16·52-s + 4·53-s − 32·68-s + 4·73-s + 3·81-s − 4·89-s − 2·97-s − 8·100-s + 2·101-s − 2·109-s − 8·117-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 8·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4·4-s − 2·9-s + 4·13-s + 6·16-s − 8·17-s − 2·25-s − 8·36-s + 2·37-s + 16·52-s + 4·53-s − 32·68-s + 4·73-s + 3·81-s − 4·89-s − 2·97-s − 8·100-s + 2·101-s − 2·109-s − 8·117-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 8·148-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 241^{16}\)
Sign: $1$
Analytic conductor: \(8.23657\times 10^{-6}\)
Root analytic conductor: \(0.693612\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 241^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5967103945\)
\(L(\frac12)\) \(\approx\) \(0.5967103945\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{4} \)
241 \( ( 1 + T )^{16} \)
good3 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
5 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
11 \( ( 1 - T^{4} + T^{8} )^{4} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
17 \( ( 1 + T + T^{2} )^{8}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
19 \( ( 1 - T^{4} + T^{8} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
31 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
53 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
59 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
71 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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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

−2.70558223641388286106719573938, −2.69319279490381370251942516586, −2.66632131201813639321400064538, −2.61365011045655484244494071303, −2.60748673432425975524900719348, −2.54735550124517164860277319875, −2.48028870742304082874394113766, −2.41805860071099034265798659020, −2.24876865051149137191498878156, −2.23183494954053702669644994386, −2.20449974343680745449640673403, −2.04296856902366493277687856296, −1.99392308413627855383003056863, −1.90409999956296988253479006873, −1.80408622068030872765819522770, −1.78411530607132770656070691459, −1.66485929761119133536384797083, −1.45269497930259395368550556650, −1.41319819404841582937990496285, −1.41189697597664661683715542095, −1.28806023641098951052220919620, −1.10951716594762928976552300407, −0.939665063057035799549020917762, −0.809778011778494890801365744751, −0.42626047175224010861918400919, 0.42626047175224010861918400919, 0.809778011778494890801365744751, 0.939665063057035799549020917762, 1.10951716594762928976552300407, 1.28806023641098951052220919620, 1.41189697597664661683715542095, 1.41319819404841582937990496285, 1.45269497930259395368550556650, 1.66485929761119133536384797083, 1.78411530607132770656070691459, 1.80408622068030872765819522770, 1.90409999956296988253479006873, 1.99392308413627855383003056863, 2.04296856902366493277687856296, 2.20449974343680745449640673403, 2.23183494954053702669644994386, 2.24876865051149137191498878156, 2.41805860071099034265798659020, 2.48028870742304082874394113766, 2.54735550124517164860277319875, 2.60748673432425975524900719348, 2.61365011045655484244494071303, 2.66632131201813639321400064538, 2.69319279490381370251942516586, 2.70558223641388286106719573938

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

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