# Properties

 Label 2-69-1.1-c5-0-5 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 − 9.32·2-s + 9·3-s + 54.9·4-s + 99.7·5-s − 83.9·6-s + 125.·7-s − 214.·8-s + 81·9-s − 930.·10-s + 177.·11-s + 494.·12-s − 919.·13-s − 1.17e3·14-s + 897.·15-s + 240.·16-s + 1.56e3·17-s − 755.·18-s + 447.·19-s + 5.48e3·20-s + 1.13e3·21-s − 1.65e3·22-s − 529·23-s − 1.93e3·24-s + 6.81e3·25-s + 8.57e3·26-s + 729·27-s + 6.92e3·28-s + ⋯
 L(s)  = 1 − 1.64·2-s + 0.577·3-s + 1.71·4-s + 1.78·5-s − 0.951·6-s + 0.971·7-s − 1.18·8-s + 0.333·9-s − 2.94·10-s + 0.442·11-s + 0.992·12-s − 1.50·13-s − 1.60·14-s + 1.02·15-s + 0.235·16-s + 1.31·17-s − 0.549·18-s + 0.284·19-s + 3.06·20-s + 0.561·21-s − 0.730·22-s − 0.208·23-s − 0.684·24-s + 2.18·25-s + 2.48·26-s + 0.192·27-s + 1.67·28-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

 $$L(3)$$ $$\approx$$ $$1.460469041$$ $$L(\frac12)$$ $$\approx$$ $$1.460469041$$ $$L(\frac{7}{2})$$ 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 + 9.32T + 32T^{2}$$
5 $$1 - 99.7T + 3.12e3T^{2}$$
7 $$1 - 125.T + 1.68e4T^{2}$$
11 $$1 - 177.T + 1.61e5T^{2}$$
13 $$1 + 919.T + 3.71e5T^{2}$$
17 $$1 - 1.56e3T + 1.41e6T^{2}$$
19 $$1 - 447.T + 2.47e6T^{2}$$
29 $$1 + 1.29e3T + 2.05e7T^{2}$$
31 $$1 + 6.65e3T + 2.86e7T^{2}$$
37 $$1 + 8.05e3T + 6.93e7T^{2}$$
41 $$1 - 1.79e4T + 1.15e8T^{2}$$
43 $$1 - 1.07e4T + 1.47e8T^{2}$$
47 $$1 + 1.78e4T + 2.29e8T^{2}$$
53 $$1 + 2.03e4T + 4.18e8T^{2}$$
59 $$1 - 2.84e4T + 7.14e8T^{2}$$
61 $$1 - 1.62e4T + 8.44e8T^{2}$$
67 $$1 - 5.41e4T + 1.35e9T^{2}$$
71 $$1 - 4.25e4T + 1.80e9T^{2}$$
73 $$1 - 4.09e4T + 2.07e9T^{2}$$
79 $$1 + 2.47e4T + 3.07e9T^{2}$$
83 $$1 + 3.08e4T + 3.93e9T^{2}$$
89 $$1 + 6.65e4T + 5.58e9T^{2}$$
97 $$1 + 1.04e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$