# Properties

 Label 2-637-91.23-c1-0-35 Degree $2$ Conductor $637$ Sign $-0.617 - 0.786i$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.73i·2-s + (−1 + 1.73i)3-s − 0.999·4-s + (−1.5 − 0.866i)5-s + (2.99 + 1.73i)6-s − 1.73i·8-s + (−0.499 − 0.866i)9-s + (−1.49 + 2.59i)10-s + (0.999 − 1.73i)12-s + (−2.5 + 2.59i)13-s + (3 − 1.73i)15-s − 5·16-s − 3·17-s + (−1.49 + 0.866i)18-s + (3 − 1.73i)19-s + (1.49 + 0.866i)20-s + ⋯
 L(s)  = 1 − 1.22i·2-s + (−0.577 + 0.999i)3-s − 0.499·4-s + (−0.670 − 0.387i)5-s + (1.22 + 0.707i)6-s − 0.612i·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s + (0.288 − 0.499i)12-s + (−0.693 + 0.720i)13-s + (0.774 − 0.447i)15-s − 1.25·16-s − 0.727·17-s + (−0.353 + 0.204i)18-s + (0.688 − 0.397i)19-s + (0.335 + 0.193i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.617 - 0.786i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (569, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.617 - 0.786i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + (2.5 - 2.59i)T$$
good2 $$1 + 1.73iT - 2T^{2}$$
3 $$1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (5.5 + 9.52i)T^{2}$$
17 $$1 + 3T + 17T^{2}$$
19 $$1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + 6T + 23T^{2}$$
29 $$1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 - 8.66iT - 37T^{2}$$
41 $$1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (3 + 1.73i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 6.92iT - 59T^{2}$$
61 $$1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 13.8iT - 83T^{2}$$
89 $$1 + 6.92iT - 89T^{2}$$
97 $$1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$