# Properties

 Label 2-1148-287.223-c0-0-1 Degree $2$ Conductor $1148$ Sign $-0.505 + 0.862i$ Analytic cond. $0.572926$ Root an. cond. $0.756919$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s + (−0.587 − 0.809i)5-s + (0.309 − 0.951i)7-s + (0.5 + 0.363i)11-s + (−0.951 + 0.309i)13-s + (−0.809 + 0.587i)15-s + (0.363 − 0.5i)17-s + (0.951 + 0.309i)19-s + (−0.951 − 0.309i)21-s + (−0.5 − 1.53i)23-s − i·27-s + (−1.30 + 0.951i)29-s + (−0.951 + 1.30i)31-s + (0.363 − 0.5i)33-s + (−0.951 + 0.309i)35-s + ⋯
 L(s)  = 1 − i·3-s + (−0.587 − 0.809i)5-s + (0.309 − 0.951i)7-s + (0.5 + 0.363i)11-s + (−0.951 + 0.309i)13-s + (−0.809 + 0.587i)15-s + (0.363 − 0.5i)17-s + (0.951 + 0.309i)19-s + (−0.951 − 0.309i)21-s + (−0.5 − 1.53i)23-s − i·27-s + (−1.30 + 0.951i)29-s + (−0.951 + 1.30i)31-s + (0.363 − 0.5i)33-s + (−0.951 + 0.309i)35-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1148$$    =    $$2^{2} \cdot 7 \cdot 41$$ Sign: $-0.505 + 0.862i$ Analytic conductor: $$0.572926$$ Root analytic conductor: $$0.756919$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1148} (797, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1148,\ (\ :0),\ -0.505 + 0.862i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9573646862$$ $$L(\frac12)$$ $$\approx$$ $$0.9573646862$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (-0.309 + 0.951i)T$$
41 $$1 + (0.951 + 0.309i)T$$
good3 $$1 + iT - T^{2}$$
5 $$1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2}$$
11 $$1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}$$
13 $$1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}$$
17 $$1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}$$
19 $$1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}$$
23 $$1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}$$
29 $$1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}$$
31 $$1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2}$$
37 $$1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}$$
43 $$1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}$$
47 $$1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}$$
53 $$1 + (0.309 - 0.951i)T^{2}$$
59 $$1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}$$
61 $$1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2}$$
67 $$1 + (0.309 - 0.951i)T^{2}$$
71 $$1 + (0.309 + 0.951i)T^{2}$$
73 $$1 + 0.618iT - T^{2}$$
79 $$1 - T + T^{2}$$
83 $$1 - iT - T^{2}$$
89 $$1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}$$
97 $$1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{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

−9.733168141825598783917784484550, −8.718705390508558449767906579069, −7.951468554396047307883152262429, −7.06252539669263118385714325967, −6.94807731915831566472753269104, −5.32398952590986787214099128300, −4.55199671937395314639290914545, −3.63110088906770882665329424562, −1.98748383566477736721312006568, −0.913343631389713829830884171142, 2.12432984687920650746321530622, 3.44922150668446790820367215838, 3.92701632411256822637741222932, 5.29979501293574239769257299898, 5.73170310685823106109399453819, 7.23837266657802459782848244315, 7.64199569956829276580802875132, 8.861825623103934583773289478759, 9.567867206998146406909420976364, 10.11461546199930615980998928838