# Properties

 Label 32-1020e16-1.1-c0e16-0-0 Degree $32$ Conductor $1.373\times 10^{48}$ Sign $1$ Analytic cond. $2.03290\times 10^{-5}$ Root an. cond. $0.713474$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## 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^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{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 3^{16} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16}$$ Sign: $1$ Analytic conductor: $$2.03290\times 10^{-5}$$ Root analytic conductor: $$0.713474$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1020} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1269532336$$ $$L(\frac12)$$ $$\approx$$ $$0.1269532336$$ $$L(1)$$ 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 + T^{16}$$
5 $$1 + T^{16}$$
17 $$1 + T^{16}$$
good7 $$( 1 + T^{16} )^{2}$$
11 $$( 1 + T^{16} )^{2}$$
13 $$( 1 + T^{4} )^{8}$$
19 $$( 1 + T^{2} )^{8}( 1 + T^{4} )^{4}$$
23 $$( 1 + T^{16} )^{2}$$
29 $$( 1 + T^{16} )^{2}$$
31 $$( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}$$
37 $$( 1 + T^{16} )^{2}$$
41 $$( 1 + T^{16} )^{2}$$
43 $$( 1 + T^{8} )^{4}$$
47 $$( 1 + T^{16} )^{2}$$
53 $$( 1 + T^{16} )^{2}$$
59 $$( 1 + T^{8} )^{4}$$
61 $$( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}$$
67 $$( 1 - T )^{16}( 1 + T )^{16}$$
71 $$( 1 + T^{16} )^{2}$$
73 $$( 1 + T^{16} )^{2}$$
79 $$( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}$$
83 $$( 1 + T^{16} )^{2}$$
89 $$( 1 + T^{4} )^{8}$$
97 $$( 1 + T^{16} )^{2}$$
show less
$$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

−2.81110309351978217735781622570, −2.80570627200465257618672620433, −2.58260026392467077265575628889, −2.53754969307520753550051482809, −2.53168305925684701834046944554, −2.43412095366924676402637839236, −2.41897542541334512804867164128, −2.40579435070972329083106504992, −2.26724841138352773518928903509, −2.20292262920009732387955782383, −2.18181223939938870383164631483, −2.12957255821904708995357952197, −2.01165920898450086482298397064, −1.79193683910883543721680266992, −1.62994738404096938322916247199, −1.55763646793779086736871781000, −1.48671161032624166855222918889, −1.29793675977589172519717675666, −1.25255579459130105101455211483, −1.19101894754925677556364277854, −1.16327607194070429750335500061, −1.15961611182617140704787241528, −1.12030872322884848244546111926, −0.848533627719318610801011031217, −0.21669656063834619983073197022, 0.21669656063834619983073197022, 0.848533627719318610801011031217, 1.12030872322884848244546111926, 1.15961611182617140704787241528, 1.16327607194070429750335500061, 1.19101894754925677556364277854, 1.25255579459130105101455211483, 1.29793675977589172519717675666, 1.48671161032624166855222918889, 1.55763646793779086736871781000, 1.62994738404096938322916247199, 1.79193683910883543721680266992, 2.01165920898450086482298397064, 2.12957255821904708995357952197, 2.18181223939938870383164631483, 2.20292262920009732387955782383, 2.26724841138352773518928903509, 2.40579435070972329083106504992, 2.41897542541334512804867164128, 2.43412095366924676402637839236, 2.53168305925684701834046944554, 2.53754969307520753550051482809, 2.58260026392467077265575628889, 2.80570627200465257618672620433, 2.81110309351978217735781622570

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

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