# Properties

 Label 2-160-5.4-c1-0-1 Degree $2$ Conductor $160$ Sign $-i$ Analytic cond. $1.27760$ Root an. cond. $1.13031$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3.23i·3-s + 2.23·5-s − 0.763i·7-s − 7.47·9-s + 7.23i·15-s + 2.47·21-s − 5.70i·23-s + 5.00·25-s − 14.4i·27-s + 6·29-s − 1.70i·35-s − 4.47·41-s + 11.2i·43-s − 16.7·45-s − 13.7i·47-s + ⋯
 L(s)  = 1 + 1.86i·3-s + 0.999·5-s − 0.288i·7-s − 2.49·9-s + 1.86i·15-s + 0.539·21-s − 1.19i·23-s + 1.00·25-s − 2.78i·27-s + 1.11·29-s − 0.288i·35-s − 0.698·41-s + 1.71i·43-s − 2.49·45-s − 1.99i·47-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$160$$    =    $$2^{5} \cdot 5$$ Sign: $-i$ Analytic conductor: $$1.27760$$ Root analytic conductor: $$1.13031$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{160} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 160,\ (\ :1/2),\ -i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.876304 + 0.876304i$$ $$L(\frac12)$$ $$\approx$$ $$0.876304 + 0.876304i$$ $$L(\frac{3}{2})$$ 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 - 2.23T$$
good3 $$1 - 3.23iT - 3T^{2}$$
7 $$1 + 0.763iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 5.70iT - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 4.47T + 41T^{2}$$
43 $$1 - 11.2iT - 43T^{2}$$
47 $$1 + 13.7iT - 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 13.4T + 61T^{2}$$
67 $$1 - 8.18iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 17.7iT - 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 - 97T^{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

−13.39568731903387427016449487468, −11.91169353785639270521400159309, −10.64189575891606623841671700192, −10.23370231809998942513444693926, −9.300931496355539349042987080767, −8.439039713800605599546754603685, −6.43519184526866561268525595499, −5.27231567091486983092416864152, −4.32262036432335171751972215288, −2.86185531446723818518215656358, 1.49656336476859230117027825145, 2.75334185645567540086841645499, 5.43352400992124616358714228380, 6.31002618409748606745822405705, 7.24428262020010073229159694421, 8.348622798880760706844654204351, 9.376028053791290275288534376888, 10.83802470521550071890652379357, 12.02634511974664323094929292530, 12.64281248849550552870984957719