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Properties

Label 32-30e32-1.1-c0e16-0-0
Degree $32$
Conductor $1.853\times 10^{47}$
Sign $1$
Analytic cond. $2.74406\times 10^{-6}$
Root an. cond. $0.670192$
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  + 4·13-s + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·13-s + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(2.74406\times 10^{-6}\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6673443423\)
\(L(\frac12)\) \(\approx\) \(0.6673443423\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( 1 \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
good7 \( ( 1 + T^{4} )^{8} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
41 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
43 \( ( 1 + T^{4} )^{8} \)
47 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
53 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
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 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
89 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−2.91839267484436708222363203051, −2.91594584388946809654208969471, −2.77181975301344430612239502277, −2.72936958872678141768368274003, −2.61660621865662822810301276553, −2.48451853843052787941143155655, −2.42427285718060389579826026736, −2.37899446072060370362761870870, −2.35682991258214264900688641940, −2.33432413161876626903061854070, −2.17929036591852459450067450580, −2.10943664465414073280822298894, −1.94411884950686831028261501544, −1.73339998890753683225415096372, −1.72699768750245405375806757333, −1.57849827473773916319013860730, −1.52614360141319530145078660978, −1.46904284801375981271767337577, −1.39622071123164531093632587568, −1.25331886107592805545477836199, −1.13257137602631963512333857783, −0.998881365348784198240016721766, −0.968637266781333851888623831607, −0.968469686603993211034802268358, −0.800737410943779763670657227828, 0.800737410943779763670657227828, 0.968469686603993211034802268358, 0.968637266781333851888623831607, 0.998881365348784198240016721766, 1.13257137602631963512333857783, 1.25331886107592805545477836199, 1.39622071123164531093632587568, 1.46904284801375981271767337577, 1.52614360141319530145078660978, 1.57849827473773916319013860730, 1.72699768750245405375806757333, 1.73339998890753683225415096372, 1.94411884950686831028261501544, 2.10943664465414073280822298894, 2.17929036591852459450067450580, 2.33432413161876626903061854070, 2.35682991258214264900688641940, 2.37899446072060370362761870870, 2.42427285718060389579826026736, 2.48451853843052787941143155655, 2.61660621865662822810301276553, 2.72936958872678141768368274003, 2.77181975301344430612239502277, 2.91594584388946809654208969471, 2.91839267484436708222363203051

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

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