# Properties

 Label 2-637-1.1-c1-0-3 Degree $2$ Conductor $637$ Sign $1$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.195·2-s + 0.259·3-s − 1.96·4-s − 3.93·5-s − 0.0508·6-s + 0.775·8-s − 2.93·9-s + 0.769·10-s + 4.50·11-s − 0.509·12-s − 13-s − 1.02·15-s + 3.77·16-s + 2.28·17-s + 0.573·18-s + 1.78·19-s + 7.71·20-s − 0.881·22-s + 1.74·23-s + 0.201·24-s + 10.4·25-s + 0.195·26-s − 1.54·27-s + 1.65·29-s + 0.199·30-s − 5.60·31-s − 2.28·32-s + ⋯
 L(s)  = 1 − 0.138·2-s + 0.149·3-s − 0.980·4-s − 1.75·5-s − 0.0207·6-s + 0.274·8-s − 0.977·9-s + 0.243·10-s + 1.35·11-s − 0.147·12-s − 0.277·13-s − 0.263·15-s + 0.942·16-s + 0.553·17-s + 0.135·18-s + 0.410·19-s + 1.72·20-s − 0.187·22-s + 0.362·23-s + 0.0411·24-s + 2.09·25-s + 0.0383·26-s − 0.296·27-s + 0.306·29-s + 0.0364·30-s − 1.00·31-s − 0.404·32-s + ⋯

## Functional equation

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

## Invariants

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

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7345528446$$ $$L(\frac12)$$ $$\approx$$ $$0.7345528446$$ $$L(\frac{3}{2})$$ 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 + T$$
good2 $$1 + 0.195T + 2T^{2}$$
3 $$1 - 0.259T + 3T^{2}$$
5 $$1 + 3.93T + 5T^{2}$$
11 $$1 - 4.50T + 11T^{2}$$
17 $$1 - 2.28T + 17T^{2}$$
19 $$1 - 1.78T + 19T^{2}$$
23 $$1 - 1.74T + 23T^{2}$$
29 $$1 - 1.65T + 29T^{2}$$
31 $$1 + 5.60T + 31T^{2}$$
37 $$1 - 7.14T + 37T^{2}$$
41 $$1 - 8.11T + 41T^{2}$$
43 $$1 - 6.81T + 43T^{2}$$
47 $$1 + 3.54T + 47T^{2}$$
53 $$1 - 3.28T + 53T^{2}$$
59 $$1 + 4.50T + 59T^{2}$$
61 $$1 + 7.54T + 61T^{2}$$
67 $$1 + 12.6T + 67T^{2}$$
71 $$1 - 9.54T + 71T^{2}$$
73 $$1 + 1.08T + 73T^{2}$$
79 $$1 - 0.791T + 79T^{2}$$
83 $$1 - 7.14T + 83T^{2}$$
89 $$1 - 11.2T + 89T^{2}$$
97 $$1 - 8.81T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$