# Properties

 Label 2-966-1.1-c3-0-0 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 − 12.7·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 25.5·10-s − 4.56·11-s − 12·12-s − 70.6·13-s + 14·14-s + 38.3·15-s + 16·16-s − 78.3·17-s − 18·18-s − 43.1·19-s − 51.1·20-s + 21·21-s + 9.12·22-s − 23·23-s + 24·24-s + 38.3·25-s + 141.·26-s − 27·27-s − 28·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.808·10-s − 0.125·11-s − 0.288·12-s − 1.50·13-s + 0.267·14-s + 0.660·15-s + 0.250·16-s − 1.11·17-s − 0.235·18-s − 0.521·19-s − 0.571·20-s + 0.218·21-s + 0.0884·22-s − 0.208·23-s + 0.204·24-s + 0.307·25-s + 1.06·26-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

 $$L(2)$$ $$\approx$$ $$0.03501366118$$ $$L(\frac12)$$ $$\approx$$ $$0.03501366118$$ $$L(\frac{5}{2})$$ 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 + 12.7T + 125T^{2}$$
11 $$1 + 4.56T + 1.33e3T^{2}$$
13 $$1 + 70.6T + 2.19e3T^{2}$$
17 $$1 + 78.3T + 4.91e3T^{2}$$
19 $$1 + 43.1T + 6.85e3T^{2}$$
29 $$1 - 166.T + 2.43e4T^{2}$$
31 $$1 + 147.T + 2.97e4T^{2}$$
37 $$1 + 363.T + 5.06e4T^{2}$$
41 $$1 + 295.T + 6.89e4T^{2}$$
43 $$1 + 456.T + 7.95e4T^{2}$$
47 $$1 + 117.T + 1.03e5T^{2}$$
53 $$1 + 221.T + 1.48e5T^{2}$$
59 $$1 + 22.7T + 2.05e5T^{2}$$
61 $$1 + 399.T + 2.26e5T^{2}$$
67 $$1 - 446.T + 3.00e5T^{2}$$
71 $$1 - 321.T + 3.57e5T^{2}$$
73 $$1 + 519.T + 3.89e5T^{2}$$
79 $$1 - 85.1T + 4.93e5T^{2}$$
83 $$1 + 163.T + 5.71e5T^{2}$$
89 $$1 + 955.T + 7.04e5T^{2}$$
97 $$1 + 476.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$