# Properties

 Degree $2$ Conductor $1997$ Sign $-1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.76·2-s − 4.70·3-s − 0.381·4-s − 6.86·5-s + 12.9·6-s + 2.49·7-s + 23.1·8-s − 4.86·9-s + 18.9·10-s − 15.4·11-s + 1.79·12-s + 54.8·13-s − 6.89·14-s + 32.3·15-s − 60.8·16-s − 123.·17-s + 13.4·18-s − 0.288·19-s + 2.61·20-s − 11.7·21-s + 42.6·22-s + 33.4·23-s − 108.·24-s − 77.8·25-s − 151.·26-s + 149.·27-s − 0.953·28-s + ⋯
 L(s)  = 1 − 0.975·2-s − 0.905·3-s − 0.0476·4-s − 0.614·5-s + 0.883·6-s + 0.134·7-s + 1.02·8-s − 0.180·9-s + 0.599·10-s − 0.423·11-s + 0.0431·12-s + 1.17·13-s − 0.131·14-s + 0.556·15-s − 0.950·16-s − 1.75·17-s + 0.175·18-s − 0.00348·19-s + 0.0292·20-s − 0.122·21-s + 0.413·22-s + 0.303·23-s − 0.925·24-s − 0.622·25-s − 1.14·26-s + 1.06·27-s − 0.00643·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1997 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1997$$ Sign: $-1$ Motivic weight: $$3$$ Character: $\chi_{1997} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1997,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad1997 $$1 - 1.99e3T$$
good2 $$1 + 2.76T + 8T^{2}$$
3 $$1 + 4.70T + 27T^{2}$$
5 $$1 + 6.86T + 125T^{2}$$
7 $$1 - 2.49T + 343T^{2}$$
11 $$1 + 15.4T + 1.33e3T^{2}$$
13 $$1 - 54.8T + 2.19e3T^{2}$$
17 $$1 + 123.T + 4.91e3T^{2}$$
19 $$1 + 0.288T + 6.85e3T^{2}$$
23 $$1 - 33.4T + 1.21e4T^{2}$$
29 $$1 - 31.0T + 2.43e4T^{2}$$
31 $$1 - 303.T + 2.97e4T^{2}$$
37 $$1 + 339.T + 5.06e4T^{2}$$
41 $$1 + 224.T + 6.89e4T^{2}$$
43 $$1 + 176.T + 7.95e4T^{2}$$
47 $$1 + 79.8T + 1.03e5T^{2}$$
53 $$1 - 528.T + 1.48e5T^{2}$$
59 $$1 + 67.2T + 2.05e5T^{2}$$
61 $$1 + 545.T + 2.26e5T^{2}$$
67 $$1 - 422.T + 3.00e5T^{2}$$
71 $$1 - 948.T + 3.57e5T^{2}$$
73 $$1 + 910.T + 3.89e5T^{2}$$
79 $$1 - 906.T + 4.93e5T^{2}$$
83 $$1 - 845.T + 5.71e5T^{2}$$
89 $$1 - 592.T + 7.04e5T^{2}$$
97 $$1 + 707.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$