# Properties

 Label 2-637-91.4-c1-0-40 Degree $2$ Conductor $637$ Sign $-0.372 - 0.927i$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.73i·2-s + (−0.5 − 0.866i)3-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (−1.49 + 0.866i)6-s − 1.73i·8-s + (1 − 1.73i)9-s + (1.49 + 2.59i)10-s + (−4.5 + 2.59i)11-s + (0.499 + 0.866i)12-s + (1 − 3.46i)13-s + (1.5 + 0.866i)15-s − 5·16-s − 6·17-s + (−3 − 1.73i)18-s + (1.5 + 0.866i)19-s + ⋯
 L(s)  = 1 − 1.22i·2-s + (−0.288 − 0.499i)3-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (−0.612 + 0.353i)6-s − 0.612i·8-s + (0.333 − 0.577i)9-s + (0.474 + 0.821i)10-s + (−1.35 + 0.783i)11-s + (0.144 + 0.249i)12-s + (0.277 − 0.960i)13-s + (0.387 + 0.223i)15-s − 1.25·16-s − 1.45·17-s + (−0.707 − 0.408i)18-s + (0.344 + 0.198i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.372 - 0.927i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (459, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.372 - 0.927i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.267371 + 0.395611i$$ $$L(\frac12)$$ $$\approx$$ $$0.267371 + 0.395611i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
13 $$1 + (-1 + 3.46i)T$$
good2 $$1 + 1.73iT - 2T^{2}$$
3 $$1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + 3.46iT - 59T^{2}$$
61 $$1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 3.46iT - 83T^{2}$$
89 $$1 - 6.92iT - 89T^{2}$$
97 $$1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$