# Properties

 Label 2-966-1.1-c3-0-12 Degree $2$ Conductor $966$ Sign $1$ Analytic cond. $56.9958$ Root an. cond. $7.54955$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s − 3·3-s + 4·4-s − 9.55·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 19.1·10-s − 35.9·11-s − 12·12-s + 6.86·13-s + 14·14-s + 28.6·15-s + 16·16-s − 16.4·17-s + 18·18-s − 27.6·19-s − 38.2·20-s − 21·21-s − 71.8·22-s − 23·23-s − 24·24-s − 33.7·25-s + 13.7·26-s − 27·27-s + 28·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.854·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.604·10-s − 0.985·11-s − 0.288·12-s + 0.146·13-s + 0.267·14-s + 0.493·15-s + 0.250·16-s − 0.235·17-s + 0.235·18-s − 0.333·19-s − 0.427·20-s − 0.218·21-s − 0.696·22-s − 0.208·23-s − 0.204·24-s − 0.270·25-s + 0.103·26-s − 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$966$$    =    $$2 \cdot 3 \cdot 7 \cdot 23$$ Sign: $1$ Analytic conductor: $$56.9958$$ Root analytic conductor: $$7.54955$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 966,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.004272684$$ $$L(\frac12)$$ $$\approx$$ $$2.004272684$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2T$$
3 $$1 + 3T$$
7 $$1 - 7T$$
23 $$1 + 23T$$
good5 $$1 + 9.55T + 125T^{2}$$
11 $$1 + 35.9T + 1.33e3T^{2}$$
13 $$1 - 6.86T + 2.19e3T^{2}$$
17 $$1 + 16.4T + 4.91e3T^{2}$$
19 $$1 + 27.6T + 6.85e3T^{2}$$
29 $$1 - 307.T + 2.43e4T^{2}$$
31 $$1 - 278.T + 2.97e4T^{2}$$
37 $$1 + 225.T + 5.06e4T^{2}$$
41 $$1 - 151.T + 6.89e4T^{2}$$
43 $$1 + 434.T + 7.95e4T^{2}$$
47 $$1 - 281.T + 1.03e5T^{2}$$
53 $$1 + 337.T + 1.48e5T^{2}$$
59 $$1 - 744.T + 2.05e5T^{2}$$
61 $$1 + 345.T + 2.26e5T^{2}$$
67 $$1 - 465.T + 3.00e5T^{2}$$
71 $$1 - 956.T + 3.57e5T^{2}$$
73 $$1 - 536.T + 3.89e5T^{2}$$
79 $$1 + 541.T + 4.93e5T^{2}$$
83 $$1 - 352.T + 5.71e5T^{2}$$
89 $$1 - 1.53e3T + 7.04e5T^{2}$$
97 $$1 + 226.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$