# Properties

 Label 2-825-1.1-c5-0-82 Degree $2$ Conductor $825$ Sign $1$ Analytic cond. $132.316$ Root an. cond. $11.5028$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.886·2-s + 9·3-s − 31.2·4-s − 7.97·6-s + 224.·7-s + 56.0·8-s + 81·9-s + 121·11-s − 280.·12-s + 1.06e3·13-s − 199.·14-s + 949.·16-s − 1.34e3·17-s − 71.8·18-s + 691.·19-s + 2.02e3·21-s − 107.·22-s + 3.39e3·23-s + 504.·24-s − 944.·26-s + 729·27-s − 7.01e3·28-s + 8.60e3·29-s − 320.·31-s − 2.63e3·32-s + 1.08e3·33-s + 1.19e3·34-s + ⋯
 L(s)  = 1 − 0.156·2-s + 0.577·3-s − 0.975·4-s − 0.0904·6-s + 1.73·7-s + 0.309·8-s + 0.333·9-s + 0.301·11-s − 0.563·12-s + 1.74·13-s − 0.271·14-s + 0.926·16-s − 1.13·17-s − 0.0522·18-s + 0.439·19-s + 1.00·21-s − 0.0472·22-s + 1.33·23-s + 0.178·24-s − 0.274·26-s + 0.192·27-s − 1.69·28-s + 1.89·29-s − 0.0599·31-s − 0.454·32-s + 0.174·33-s + 0.177·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$825$$    =    $$3 \cdot 5^{2} \cdot 11$$ Sign: $1$ Analytic conductor: $$132.316$$ Root analytic conductor: $$11.5028$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 825,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$3.299564746$$ $$L(\frac12)$$ $$\approx$$ $$3.299564746$$ $$L(\frac{7}{2})$$ 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$$
5 $$1$$
11 $$1 - 121T$$
good2 $$1 + 0.886T + 32T^{2}$$
7 $$1 - 224.T + 1.68e4T^{2}$$
13 $$1 - 1.06e3T + 3.71e5T^{2}$$
17 $$1 + 1.34e3T + 1.41e6T^{2}$$
19 $$1 - 691.T + 2.47e6T^{2}$$
23 $$1 - 3.39e3T + 6.43e6T^{2}$$
29 $$1 - 8.60e3T + 2.05e7T^{2}$$
31 $$1 + 320.T + 2.86e7T^{2}$$
37 $$1 + 1.90e3T + 6.93e7T^{2}$$
41 $$1 - 5.82e3T + 1.15e8T^{2}$$
43 $$1 - 1.42e3T + 1.47e8T^{2}$$
47 $$1 + 6.45e3T + 2.29e8T^{2}$$
53 $$1 - 9.10e3T + 4.18e8T^{2}$$
59 $$1 - 1.02e4T + 7.14e8T^{2}$$
61 $$1 + 3.42e4T + 8.44e8T^{2}$$
67 $$1 + 6.84e4T + 1.35e9T^{2}$$
71 $$1 - 1.88e3T + 1.80e9T^{2}$$
73 $$1 + 8.77e4T + 2.07e9T^{2}$$
79 $$1 + 5.24e4T + 3.07e9T^{2}$$
83 $$1 - 5.86e4T + 3.93e9T^{2}$$
89 $$1 + 1.13e5T + 5.58e9T^{2}$$
97 $$1 + 1.15e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$