Properties

Label 32-3744e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.491\times 10^{57}$
Sign $1$
Analytic cond. $22074.3$
Root an. cond. $1.36693$
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  + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 263-s + 269-s + ⋯
L(s)  = 1  + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 263-s + 269-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{80} \cdot 3^{32} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(22074.3\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3744} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{80} \cdot 3^{32} \cdot 13^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8236006283\)
\(L(\frac12)\) \(\approx\) \(0.8236006283\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{16} \)
3 \( 1 \)
13 \( ( 1 + T^{8} )^{2} \)
good5 \( ( 1 + T^{16} )^{2} \)
7 \( ( 1 + T^{4} )^{8} \)
11 \( ( 1 + T^{16} )^{2} \)
17 \( ( 1 - T )^{16}( 1 + T )^{16} \)
19 \( ( 1 + T^{8} )^{4} \)
23 \( ( 1 + T^{4} )^{8} \)
29 \( ( 1 + T^{8} )^{4} \)
31 \( ( 1 + T^{2} )^{16} \)
37 \( ( 1 + T^{8} )^{4} \)
41 \( ( 1 + T^{16} )^{2} \)
43 \( ( 1 + T^{2} )^{8}( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{16} )^{2} \)
53 \( ( 1 + T^{8} )^{4} \)
59 \( ( 1 + T^{16} )^{2} \)
61 \( ( 1 + T^{8} )^{4} \)
67 \( ( 1 + T^{8} )^{4} \)
71 \( ( 1 + T^{16} )^{2} \)
73 \( ( 1 + T^{4} )^{8} \)
79 \( ( 1 + T^{8} )^{4} \)
83 \( ( 1 + T^{16} )^{2} \)
89 \( ( 1 + T^{16} )^{2} \)
97 \( ( 1 - T )^{16}( 1 + T )^{16} \)
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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

−2.30624286798423139116859177947, −2.29410116457381477646665250603, −2.21619113403132724087536378468, −2.01664062260587039899368664837, −1.93858071309876101316100778621, −1.90348298689615791285213196556, −1.88925139211463666310104043789, −1.82185212663128567394793473161, −1.76831084494199667875565757354, −1.70820051307712964802692737023, −1.67403356825560934344185455718, −1.66313819059218628226822829482, −1.45174902454615745102911940181, −1.38280807987676023904890400443, −1.29555610683493295857639325008, −1.19078768396284301261183835688, −1.04510462627879012596324314407, −1.01401520019469534485339526944, −0.927971766278985219227596431932, −0.879221562805215226194782368553, −0.838049970990197333240654532713, −0.796706179152182334106899950194, −0.72447943024161130438743682236, −0.25654025324044927180753953098, −0.19295477679010845530618558627, 0.19295477679010845530618558627, 0.25654025324044927180753953098, 0.72447943024161130438743682236, 0.796706179152182334106899950194, 0.838049970990197333240654532713, 0.879221562805215226194782368553, 0.927971766278985219227596431932, 1.01401520019469534485339526944, 1.04510462627879012596324314407, 1.19078768396284301261183835688, 1.29555610683493295857639325008, 1.38280807987676023904890400443, 1.45174902454615745102911940181, 1.66313819059218628226822829482, 1.67403356825560934344185455718, 1.70820051307712964802692737023, 1.76831084494199667875565757354, 1.82185212663128567394793473161, 1.88925139211463666310104043789, 1.90348298689615791285213196556, 1.93858071309876101316100778621, 2.01664062260587039899368664837, 2.21619113403132724087536378468, 2.29410116457381477646665250603, 2.30624286798423139116859177947

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

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