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Properties

Label 32-3751e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.536\times 10^{57}$
Sign $1$
Analytic cond. $22744.0$
Root an. cond. $1.36820$
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  − 4·4-s + 2·5-s − 2·9-s + 6·16-s − 8·20-s + 16·23-s + 25-s + 2·31-s + 8·36-s − 2·37-s − 4·45-s + 4·47-s + 2·53-s − 2·59-s − 4·64-s − 8·67-s − 2·71-s + 12·80-s + 3·81-s − 16·89-s − 64·92-s − 4·97-s − 4·100-s + 2·113-s + 32·115-s − 8·124-s + 2·125-s + ⋯
L(s)  = 1  − 4·4-s + 2·5-s − 2·9-s + 6·16-s − 8·20-s + 16·23-s + 25-s + 2·31-s + 8·36-s − 2·37-s − 4·45-s + 4·47-s + 2·53-s − 2·59-s − 4·64-s − 8·67-s − 2·71-s + 12·80-s + 3·81-s − 16·89-s − 64·92-s − 4·97-s − 4·100-s + 2·113-s + 32·115-s − 8·124-s + 2·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{32} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(11^{32} \cdot 31^{16}\)
Sign: $1$
Analytic conductor: \(22744.0\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 11^{32} \cdot 31^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3162782977\)
\(L(\frac12)\) \(\approx\) \(0.3162782977\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
3 \( ( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
17 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
23 \( ( 1 - T + T^{2} )^{16} \)
29 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
61 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
67 \( ( 1 + T )^{16}( 1 - T + T^{2} )^{8} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
73 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
79 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
83 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
89 \( ( 1 + T + T^{2} )^{16} \)
97 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{4} \)
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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

−2.34778696967127634887796419492, −2.26263244930044222940584417165, −2.25860770626319611940331567529, −2.15565392735535723672778181526, −2.02709906652954179119783413409, −1.77658749321366998353438395006, −1.76025986546471248835351211616, −1.73367338877712200789060455327, −1.63433653208910694838239504267, −1.59635358815405928868837484516, −1.48485104231942077060450350554, −1.44527062964230454953726309556, −1.41237352689652520690846167747, −1.35894648925849399320454698038, −1.30235180338926152664176073569, −1.13831092630051854704886437056, −1.10164912889388455584827249248, −1.02440405677036609404236154897, −0.964658968453201500635320506469, −0.915345328488327500494692280005, −0.799409036421133017419107518599, −0.77349691047938558046018726640, −0.68599610465467807361203350766, −0.25435981489947852331373513280, −0.16049044714255539052317022285, 0.16049044714255539052317022285, 0.25435981489947852331373513280, 0.68599610465467807361203350766, 0.77349691047938558046018726640, 0.799409036421133017419107518599, 0.915345328488327500494692280005, 0.964658968453201500635320506469, 1.02440405677036609404236154897, 1.10164912889388455584827249248, 1.13831092630051854704886437056, 1.30235180338926152664176073569, 1.35894648925849399320454698038, 1.41237352689652520690846167747, 1.44527062964230454953726309556, 1.48485104231942077060450350554, 1.59635358815405928868837484516, 1.63433653208910694838239504267, 1.73367338877712200789060455327, 1.76025986546471248835351211616, 1.77658749321366998353438395006, 2.02709906652954179119783413409, 2.15565392735535723672778181526, 2.25860770626319611940331567529, 2.26263244930044222940584417165, 2.34778696967127634887796419492

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

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