# Properties

 Label 2-1150-5.4-c3-0-75 Degree $2$ Conductor $1150$ Sign $-0.447 + 0.894i$ Analytic cond. $67.8521$ Root an. cond. $8.23724$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·2-s + 7.57i·3-s − 4·4-s + 15.1·6-s − 35.4i·7-s + 8i·8-s − 30.3·9-s − 16.6·11-s − 30.2i·12-s + 79.9i·13-s − 70.8·14-s + 16·16-s − 46.8i·17-s + 60.6i·18-s + 110.·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 1.45i·3-s − 0.5·4-s + 1.03·6-s − 1.91i·7-s + 0.353i·8-s − 1.12·9-s − 0.455·11-s − 0.728i·12-s + 1.70i·13-s − 1.35·14-s + 0.250·16-s − 0.667i·17-s + 0.794i·18-s + 1.33·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1150$$    =    $$2 \cdot 5^{2} \cdot 23$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$67.8521$$ Root analytic conductor: $$8.23724$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1150} (599, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1150,\ (\ :3/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.9260279340$$ $$L(\frac12)$$ $$\approx$$ $$0.9260279340$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2iT$$
5 $$1$$
23 $$1 - 23iT$$
good3 $$1 - 7.57iT - 27T^{2}$$
7 $$1 + 35.4iT - 343T^{2}$$
11 $$1 + 16.6T + 1.33e3T^{2}$$
13 $$1 - 79.9iT - 2.19e3T^{2}$$
17 $$1 + 46.8iT - 4.91e3T^{2}$$
19 $$1 - 110.T + 6.85e3T^{2}$$
29 $$1 - 0.836T + 2.43e4T^{2}$$
31 $$1 + 119.T + 2.97e4T^{2}$$
37 $$1 - 368. iT - 5.06e4T^{2}$$
41 $$1 + 95.7T + 6.89e4T^{2}$$
43 $$1 + 331. iT - 7.95e4T^{2}$$
47 $$1 + 535. iT - 1.03e5T^{2}$$
53 $$1 + 409. iT - 1.48e5T^{2}$$
59 $$1 - 352.T + 2.05e5T^{2}$$
61 $$1 + 507.T + 2.26e5T^{2}$$
67 $$1 + 820. iT - 3.00e5T^{2}$$
71 $$1 + 733.T + 3.57e5T^{2}$$
73 $$1 - 91.4iT - 3.89e5T^{2}$$
79 $$1 + 329.T + 4.93e5T^{2}$$
83 $$1 - 753. iT - 5.71e5T^{2}$$
89 $$1 - 1.05e3T + 7.04e5T^{2}$$
97 $$1 + 271. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$