# Properties

 Label 2-825-1.1-c5-0-124 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.42·2-s + 9·3-s − 2.52·4-s − 48.8·6-s + 29.0·7-s + 187.·8-s + 81·9-s + 121·11-s − 22.7·12-s + 433.·13-s − 157.·14-s − 936.·16-s − 681.·17-s − 439.·18-s + 1.40e3·19-s + 261.·21-s − 656.·22-s − 4.76e3·23-s + 1.68e3·24-s − 2.35e3·26-s + 729·27-s − 73.5·28-s − 646.·29-s + 840.·31-s − 913.·32-s + 1.08e3·33-s + 3.70e3·34-s + ⋯
 L(s)  = 1 − 0.959·2-s + 0.577·3-s − 0.0789·4-s − 0.554·6-s + 0.224·7-s + 1.03·8-s + 0.333·9-s + 0.301·11-s − 0.0456·12-s + 0.710·13-s − 0.215·14-s − 0.914·16-s − 0.572·17-s − 0.319·18-s + 0.889·19-s + 0.129·21-s − 0.289·22-s − 1.87·23-s + 0.597·24-s − 0.682·26-s + 0.192·27-s − 0.0177·28-s − 0.142·29-s + 0.157·31-s − 0.157·32-s + 0.174·33-s + 0.548·34-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: $$1$$ Selberg data: $$(2,\ 825,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 + 5.42T + 32T^{2}$$
7 $$1 - 29.0T + 1.68e4T^{2}$$
13 $$1 - 433.T + 3.71e5T^{2}$$
17 $$1 + 681.T + 1.41e6T^{2}$$
19 $$1 - 1.40e3T + 2.47e6T^{2}$$
23 $$1 + 4.76e3T + 6.43e6T^{2}$$
29 $$1 + 646.T + 2.05e7T^{2}$$
31 $$1 - 840.T + 2.86e7T^{2}$$
37 $$1 + 7.78e3T + 6.93e7T^{2}$$
41 $$1 + 1.57e4T + 1.15e8T^{2}$$
43 $$1 - 857.T + 1.47e8T^{2}$$
47 $$1 - 9.40e3T + 2.29e8T^{2}$$
53 $$1 - 3.88e4T + 4.18e8T^{2}$$
59 $$1 - 4.33e4T + 7.14e8T^{2}$$
61 $$1 + 3.56e4T + 8.44e8T^{2}$$
67 $$1 - 4.28e3T + 1.35e9T^{2}$$
71 $$1 - 790.T + 1.80e9T^{2}$$
73 $$1 + 5.43e4T + 2.07e9T^{2}$$
79 $$1 - 2.46e4T + 3.07e9T^{2}$$
83 $$1 + 6.77e4T + 3.93e9T^{2}$$
89 $$1 + 4.30e4T + 5.58e9T^{2}$$
97 $$1 - 7.72e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$