# Properties

 Label 2-275-5.4-c1-0-12 Degree $2$ Conductor $275$ Sign $-0.447 + 0.894i$ Analytic cond. $2.19588$ Root an. cond. $1.48185$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.414i·2-s − 2.82i·3-s + 1.82·4-s − 1.17·6-s − 2i·7-s − 1.58i·8-s − 5.00·9-s + 11-s − 5.17i·12-s + 6.82i·13-s − 0.828·14-s + 3·16-s + 1.17i·17-s + 2.07i·18-s − 5.65·21-s − 0.414i·22-s + ⋯
 L(s)  = 1 − 0.292i·2-s − 1.63i·3-s + 0.914·4-s − 0.478·6-s − 0.755i·7-s − 0.560i·8-s − 1.66·9-s + 0.301·11-s − 1.49i·12-s + 1.89i·13-s − 0.221·14-s + 0.750·16-s + 0.284i·17-s + 0.488i·18-s − 1.23·21-s − 0.0883i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$275$$    =    $$5^{2} \cdot 11$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$2.19588$$ Root analytic conductor: $$1.48185$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{275} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 275,\ (\ :1/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.797967 - 1.29113i$$ $$L(\frac12)$$ $$\approx$$ $$0.797967 - 1.29113i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
11 $$1 - T$$
good2 $$1 + 0.414iT - 2T^{2}$$
3 $$1 + 2.82iT - 3T^{2}$$
7 $$1 + 2iT - 7T^{2}$$
13 $$1 - 6.82iT - 13T^{2}$$
17 $$1 - 1.17iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 2.82iT - 23T^{2}$$
29 $$1 + 7.65T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 3.65iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 - 6iT - 43T^{2}$$
47 $$1 + 2.82iT - 47T^{2}$$
53 $$1 + 0.343iT - 53T^{2}$$
59 $$1 - 9.65T + 59T^{2}$$
61 $$1 - 13.3T + 61T^{2}$$
67 $$1 + 4.48iT - 67T^{2}$$
71 $$1 + 11.3T + 71T^{2}$$
73 $$1 - 6.82iT - 73T^{2}$$
79 $$1 + 4T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 + 9.31T + 89T^{2}$$
97 $$1 + 7.65iT - 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

−11.59295244178732043903599753799, −11.12360429289197452009420931951, −9.760389359797121367145887006245, −8.454979523558487280598901720878, −7.26990349252116302854572442786, −6.89395966359866780253505746893, −6.02259013673371578289011117629, −3.99854168546424015710266382145, −2.32687646222128897370739740071, −1.33719664321169824589643543215, 2.67464430784910526810956774364, 3.73815666774460920540588790787, 5.40206477674209174978974510786, 5.72171463244198737573255066967, 7.39265919229779502224473295362, 8.478838844715940628802910640840, 9.464258081518195992711099351224, 10.37280365643641391671080200478, 11.05769676662173517101600625210, 11.89764355349030160910439024529