# Properties

 Label 1-160-160.27-r0-0-0 Degree $1$ Conductor $160$ Sign $0.681 - 0.731i$ Analytic cond. $0.743036$ Root an. cond. $0.743036$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)3-s + 7-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s − i·17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s + 23-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s − i·33-s + (−0.707 + 0.707i)37-s − i·39-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)3-s + 7-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s − i·17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s + 23-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s − i·33-s + (−0.707 + 0.707i)37-s − i·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $0.681 - 0.731i$ Analytic conductor: $$0.743036$$ Root analytic conductor: $$0.743036$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{160} (27, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 160,\ (0:\ ),\ 0.681 - 0.731i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9101689850 - 0.3960599601i$$ $$L(\frac12)$$ $$\approx$$ $$0.9101689850 - 0.3960599601i$$ $$L(1)$$ $$\approx$$ $$0.9307089166 - 0.2308418525i$$ $$L(1)$$ $$\approx$$ $$0.9307089166 - 0.2308418525i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + (-0.707 - 0.707i)T$$
7 $$1 + T$$
11 $$1 + (0.707 + 0.707i)T$$
13 $$1 + (-0.707 - 0.707i)T$$
17 $$1 - iT$$
19 $$1 + (0.707 - 0.707i)T$$
23 $$1 + T$$
29 $$1 + (0.707 - 0.707i)T$$
31 $$1 - T$$
37 $$1 + (-0.707 + 0.707i)T$$
41 $$1 - iT$$
43 $$1 + (0.707 - 0.707i)T$$
47 $$1 + iT$$
53 $$1 + (-0.707 + 0.707i)T$$
59 $$1 + (0.707 + 0.707i)T$$
61 $$1 + (-0.707 + 0.707i)T$$
67 $$1 + (0.707 + 0.707i)T$$
71 $$1 - iT$$
73 $$1 - T$$
79 $$1 - T$$
83 $$1 + (0.707 + 0.707i)T$$
89 $$1 - iT$$
97 $$1 + iT$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−27.76812729835312519120152392208, −27.09916483923375163040190591859, −26.37816417484005399250202489286, −24.82352256882932393682465446086, −24.011139996471189872301310820550, −23.07495555592713017557178153200, −21.808224933879595074275047237810, −21.44590729694975573710082786102, −20.2676559661811969639506066564, −19.02905933360468029958465077724, −17.808884617789974818133173180772, −16.9798950088707752585489789392, −16.203469765259166250551083422899, −14.81401985251042175900980218141, −14.25914144294554027685091715853, −12.49998516235453340363437410930, −11.51356235816057314164650727801, −10.78943558935228141898574537798, −9.5251859056442567911175100783, −8.46664669199954949379154105112, −6.94973935488019096893675409397, −5.685494188128314379734287038845, −4.67485417028504886219423317308, −3.530773444132406594607545375852, −1.45868257978250108641184873182, 1.12370116796876730111395763647, 2.5402723979605657848810833565, 4.64167992884722649442906956267, 5.43179179565065155000769772192, 7.00101161715341240687292288546, 7.62939125895061455810439256511, 9.117904911451569125756618882871, 10.54085357515506456937353340585, 11.60601705691310793566410543032, 12.28978246799140132826021877584, 13.54557928310559914778068074063, 14.5642152528688991566300065510, 15.751854811414704242879619450066, 17.247531760160840037635618297186, 17.56902225142179235751192314212, 18.63798738165399379893945393341, 19.79550064766797918332238355053, 20.756247126382788245761102148007, 22.15778004891226362550678544665, 22.75478013566759636648183901874, 23.92237206584886463439306979317, 24.677101564265189855952110412200, 25.40249019661867909653934988382, 27.130680368759292004122128828790, 27.59533133494779176717527218787