# Properties

 Label 2-69-1.1-c5-0-6 Degree $2$ Conductor $69$ Sign $1$ Analytic cond. $11.0664$ Root an. cond. $3.32663$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.24·2-s + 9·3-s − 26.9·4-s + 53.3·5-s + 20.2·6-s + 89.8·7-s − 132.·8-s + 81·9-s + 120.·10-s + 225.·11-s − 242.·12-s + 725.·13-s + 202.·14-s + 480.·15-s + 564.·16-s + 44.9·17-s + 182.·18-s + 1.21e3·19-s − 1.43e3·20-s + 808.·21-s + 505.·22-s − 529·23-s − 1.19e3·24-s − 274.·25-s + 1.63e3·26-s + 729·27-s − 2.42e3·28-s + ⋯
 L(s)  = 1 + 0.397·2-s + 0.577·3-s − 0.842·4-s + 0.955·5-s + 0.229·6-s + 0.693·7-s − 0.732·8-s + 0.333·9-s + 0.379·10-s + 0.560·11-s − 0.486·12-s + 1.19·13-s + 0.275·14-s + 0.551·15-s + 0.551·16-s + 0.0377·17-s + 0.132·18-s + 0.770·19-s − 0.804·20-s + 0.400·21-s + 0.222·22-s − 0.208·23-s − 0.422·24-s − 0.0878·25-s + 0.473·26-s + 0.192·27-s − 0.583·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $1$ Analytic conductor: $$11.0664$$ Root analytic conductor: $$3.32663$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{69} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.724661241$$ $$L(\frac12)$$ $$\approx$$ $$2.724661241$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 9T$$
23 $$1 + 529T$$
good2 $$1 - 2.24T + 32T^{2}$$
5 $$1 - 53.3T + 3.12e3T^{2}$$
7 $$1 - 89.8T + 1.68e4T^{2}$$
11 $$1 - 225.T + 1.61e5T^{2}$$
13 $$1 - 725.T + 3.71e5T^{2}$$
17 $$1 - 44.9T + 1.41e6T^{2}$$
19 $$1 - 1.21e3T + 2.47e6T^{2}$$
29 $$1 - 2.59e3T + 2.05e7T^{2}$$
31 $$1 - 585.T + 2.86e7T^{2}$$
37 $$1 + 3.84e3T + 6.93e7T^{2}$$
41 $$1 + 4.29e3T + 1.15e8T^{2}$$
43 $$1 + 2.05e4T + 1.47e8T^{2}$$
47 $$1 + 5.22e3T + 2.29e8T^{2}$$
53 $$1 + 1.57e4T + 4.18e8T^{2}$$
59 $$1 + 1.37e4T + 7.14e8T^{2}$$
61 $$1 + 1.89e4T + 8.44e8T^{2}$$
67 $$1 - 1.95e3T + 1.35e9T^{2}$$
71 $$1 - 7.52e4T + 1.80e9T^{2}$$
73 $$1 + 4.17e3T + 2.07e9T^{2}$$
79 $$1 + 9.58e4T + 3.07e9T^{2}$$
83 $$1 - 1.33e4T + 3.93e9T^{2}$$
89 $$1 - 5.09e4T + 5.58e9T^{2}$$
97 $$1 - 4.42e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$