# Properties

 Label 32-980e16-1.1-c0e16-0-0 Degree $32$ Conductor $7.238\times 10^{47}$ Sign $1$ Analytic cond. $1.07184\times 10^{-5}$ Root an. cond. $0.699345$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2·16-s − 8·53-s + 4·81-s − 16·113-s − 8·121-s + 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 + ⋯
 L(s)  = 1 + 2·16-s − 8·53-s + 4·81-s − 16·113-s − 8·121-s + 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 + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 7^{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 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{32} \cdot 5^{16} \cdot 7^{32}$$ Sign: $1$ Analytic conductor: $$1.07184\times 10^{-5}$$ Root analytic conductor: $$0.699345$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{980} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{32} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [0]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1540889744$$ $$L(\frac12)$$ $$\approx$$ $$0.1540889744$$ $$L(1)$$ 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} )^{2}$$
5 $$1 - T^{8} + T^{16}$$
7 $$1$$
good3 $$( 1 - T^{4} + T^{8} )^{4}$$
11 $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
13 $$( 1 + T^{8} )^{4}$$
17 $$( 1 - T^{8} + T^{16} )^{2}$$
19 $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
23 $$( 1 - T^{4} + T^{8} )^{4}$$
29 $$( 1 + T^{4} )^{8}$$
31 $$( 1 - T^{2} + T^{4} )^{8}$$
37 $$( 1 - T^{4} + T^{8} )^{4}$$
41 $$( 1 + T^{8} )^{4}$$
43 $$( 1 + T^{4} )^{8}$$
47 $$( 1 - T^{4} + T^{8} )^{4}$$
53 $$( 1 + T + T^{2} )^{8}( 1 - T^{2} + T^{4} )^{4}$$
59 $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
61 $$( 1 - T^{8} + T^{16} )^{2}$$
67 $$( 1 - T^{4} + T^{8} )^{4}$$
71 $$( 1 - T )^{16}( 1 + T )^{16}$$
73 $$( 1 - T^{8} + T^{16} )^{2}$$
79 $$( 1 - T^{2} + T^{4} )^{8}$$
83 $$( 1 + T^{4} )^{8}$$
89 $$( 1 - T^{8} + T^{16} )^{2}$$
97 $$( 1 + 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

−2.90908789726601974994207346512, −2.87879930550531657527706266394, −2.81285987596997729614699326455, −2.67855138154227589989887985242, −2.57688574310364587128843940915, −2.47895911302942702865657782955, −2.38780350328171679006700652100, −2.32359592202046439146272450793, −2.31622087899353581938902121366, −2.26745984249529510201006063058, −2.25334923901345136494817088892, −1.91222798708511541510137391799, −1.90046720803874689887790378417, −1.80173767710441374026707575090, −1.70160829947482711061508608904, −1.52419860039426188853104628960, −1.43215556003379197628671877211, −1.37713985780495308752362656855, −1.35823294481175125934541260987, −1.26534666742680647653959682610, −1.25477056353931840363894589746, −1.22572677767808113077800538271, −0.966980744804699731186136184365, −0.886067886780049811381084181155, −0.25224155225769042694524968774, 0.25224155225769042694524968774, 0.886067886780049811381084181155, 0.966980744804699731186136184365, 1.22572677767808113077800538271, 1.25477056353931840363894589746, 1.26534666742680647653959682610, 1.35823294481175125934541260987, 1.37713985780495308752362656855, 1.43215556003379197628671877211, 1.52419860039426188853104628960, 1.70160829947482711061508608904, 1.80173767710441374026707575090, 1.90046720803874689887790378417, 1.91222798708511541510137391799, 2.25334923901345136494817088892, 2.26745984249529510201006063058, 2.31622087899353581938902121366, 2.32359592202046439146272450793, 2.38780350328171679006700652100, 2.47895911302942702865657782955, 2.57688574310364587128843940915, 2.67855138154227589989887985242, 2.81285987596997729614699326455, 2.87879930550531657527706266394, 2.90908789726601974994207346512

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

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