# Properties

 Label 2-65e2-1.1-c1-0-10 Degree $2$ Conductor $4225$ Sign $1$ Analytic cond. $33.7367$ Root an. cond. $5.80833$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.554·2-s − 0.801·3-s − 1.69·4-s + 0.445·6-s − 2.69·7-s + 2.04·8-s − 2.35·9-s + 1.19·11-s + 1.35·12-s + 1.49·14-s + 2.24·16-s − 1.13·17-s + 1.30·18-s − 1.93·19-s + 2.15·21-s − 0.664·22-s + 4.60·23-s − 1.64·24-s + 4.29·27-s + 4.55·28-s − 7.89·29-s − 5.89·31-s − 5.34·32-s − 0.960·33-s + 0.631·34-s + 3.98·36-s + 0.951·37-s + ⋯
 L(s)  = 1 − 0.392·2-s − 0.462·3-s − 0.846·4-s + 0.181·6-s − 1.01·7-s + 0.724·8-s − 0.785·9-s + 0.361·11-s + 0.391·12-s + 0.399·14-s + 0.561·16-s − 0.275·17-s + 0.308·18-s − 0.444·19-s + 0.471·21-s − 0.141·22-s + 0.959·23-s − 0.335·24-s + 0.826·27-s + 0.860·28-s − 1.46·29-s − 1.05·31-s − 0.944·32-s − 0.167·33-s + 0.108·34-s + 0.664·36-s + 0.156·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4225$$    =    $$5^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$33.7367$$ Root analytic conductor: $$5.80833$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4225,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3395315687$$ $$L(\frac12)$$ $$\approx$$ $$0.3395315687$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
13 $$1$$
good2 $$1 + 0.554T + 2T^{2}$$
3 $$1 + 0.801T + 3T^{2}$$
7 $$1 + 2.69T + 7T^{2}$$
11 $$1 - 1.19T + 11T^{2}$$
17 $$1 + 1.13T + 17T^{2}$$
19 $$1 + 1.93T + 19T^{2}$$
23 $$1 - 4.60T + 23T^{2}$$
29 $$1 + 7.89T + 29T^{2}$$
31 $$1 + 5.89T + 31T^{2}$$
37 $$1 - 0.951T + 37T^{2}$$
41 $$1 + 3.31T + 41T^{2}$$
43 $$1 + 7.15T + 43T^{2}$$
47 $$1 + 7.69T + 47T^{2}$$
53 $$1 + 5.87T + 53T^{2}$$
59 $$1 - 0.0120T + 59T^{2}$$
61 $$1 + 8.03T + 61T^{2}$$
67 $$1 + 9.25T + 67T^{2}$$
71 $$1 - 13.7T + 71T^{2}$$
73 $$1 - 12.8T + 73T^{2}$$
79 $$1 - 0.807T + 79T^{2}$$
83 $$1 + 16.3T + 83T^{2}$$
89 $$1 - 14.7T + 89T^{2}$$
97 $$1 - 3.13T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$