# 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 $0$

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## 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.