# Properties

 Label 2-966-1.1-c1-0-0 Degree $2$ Conductor $966$ Sign $1$ Analytic cond. $7.71354$ Root an. cond. $2.77732$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 1.56·5-s + 6-s − 7-s − 8-s + 9-s + 1.56·10-s + 5.12·11-s − 12-s − 3.56·13-s + 14-s + 1.56·15-s + 16-s − 1.12·17-s − 18-s − 5.12·19-s − 1.56·20-s + 21-s − 5.12·22-s + 23-s + 24-s − 2.56·25-s + 3.56·26-s − 27-s − 28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.698·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s + 1.54·11-s − 0.288·12-s − 0.987·13-s + 0.267·14-s + 0.403·15-s + 0.250·16-s − 0.272·17-s − 0.235·18-s − 1.17·19-s − 0.349·20-s + 0.218·21-s − 1.09·22-s + 0.208·23-s + 0.204·24-s − 0.512·25-s + 0.698·26-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(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: $$7.71354$$ Root analytic conductor: $$2.77732$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 966,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7028303445$$ $$L(\frac12)$$ $$\approx$$ $$0.7028303445$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
7 $$1 + T$$
23 $$1 - T$$
good5 $$1 + 1.56T + 5T^{2}$$
11 $$1 - 5.12T + 11T^{2}$$
13 $$1 + 3.56T + 13T^{2}$$
17 $$1 + 1.12T + 17T^{2}$$
19 $$1 + 5.12T + 19T^{2}$$
29 $$1 - 7.56T + 29T^{2}$$
31 $$1 - 3.12T + 31T^{2}$$
37 $$1 + 1.56T + 37T^{2}$$
41 $$1 - 3.56T + 41T^{2}$$
43 $$1 - 6.68T + 43T^{2}$$
47 $$1 - 2.43T + 47T^{2}$$
53 $$1 - 14.2T + 53T^{2}$$
59 $$1 - 4.87T + 59T^{2}$$
61 $$1 - 0.876T + 61T^{2}$$
67 $$1 + 1.12T + 67T^{2}$$
71 $$1 + 9.36T + 71T^{2}$$
73 $$1 - 9.12T + 73T^{2}$$
79 $$1 - 14.2T + 79T^{2}$$
83 $$1 + 9.12T + 83T^{2}$$
89 $$1 - 14T + 89T^{2}$$
97 $$1 + 12.4T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$