# Properties

 Label 2-160-5.2-c0-0-0 Degree $2$ Conductor $160$ Sign $0.973 + 0.229i$ Analytic cond. $0.0798504$ Root an. cond. $0.282578$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − i·5-s + i·9-s + (−1 + i)13-s + (−1 − i)17-s − 25-s + (1 + i)37-s + 45-s − i·49-s + (1 − i)53-s + (1 + i)65-s + (1 − i)73-s − 81-s + (−1 + i)85-s + (1 + i)97-s − 2·101-s + ⋯
 L(s)  = 1 − i·5-s + i·9-s + (−1 + i)13-s + (−1 − i)17-s − 25-s + (1 + i)37-s + 45-s − i·49-s + (1 − i)53-s + (1 + i)65-s + (1 − i)73-s − 81-s + (−1 + i)85-s + (1 + i)97-s − 2·101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $0.973 + 0.229i$ Analytic conductor: $$0.0798504$$ Root analytic conductor: $$0.282578$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{160} (97, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :0),\ 0.973 + 0.229i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6357565859$$ $$L(\frac12)$$ $$\approx$$ $$0.6357565859$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + iT$$
good3 $$1 - iT^{2}$$
7 $$1 + iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
17 $$1 + (1 + i)T + iT^{2}$$
19 $$1 - T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (-1 - i)T + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + (-1 + i)T - iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (-1 + i)T - iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 - T^{2}$$
97 $$1 + (-1 - i)T + iT^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.22226723029430935142271601052, −12.05397578143127107551365370655, −11.29170416241625925518280239708, −9.913649929430436329933245040040, −9.047836141025918656673687999356, −7.980117240652363769239370864423, −6.83939292138785200432351293434, −5.19020953810386242049906005307, −4.43966521530663007674469412826, −2.21888865321118726163530306768, 2.66147261116974021335730422679, 4.03087968401882776601714179342, 5.80140199949532992853905517256, 6.80059945319875688719833291978, 7.85863292749122721691212523124, 9.221074973098363193163920528040, 10.24071687327993332863047005091, 11.07925748798546145881732505980, 12.21353085036213570284905205661, 13.06560846434277772522137725342