# Properties

 Label 2-70e2-1.1-c1-0-31 Degree $2$ Conductor $4900$ Sign $1$ Analytic cond. $39.1266$ Root an. cond. $6.25513$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.82·3-s + 5.00·9-s + 4·11-s − 4.24·13-s − 1.41·17-s + 2.82·19-s + 4·23-s + 5.65·27-s + 8·29-s + 11.3·33-s + 8·37-s − 12·39-s − 7.07·41-s + 4·43-s − 5.65·47-s − 4.00·51-s − 10·53-s + 8.00·57-s + 14.1·59-s − 7.07·61-s + 11.3·69-s + 7.07·73-s + 8·79-s + 1.00·81-s + 14.1·83-s + 22.6·87-s + 7.07·89-s + ⋯
 L(s)  = 1 + 1.63·3-s + 1.66·9-s + 1.20·11-s − 1.17·13-s − 0.342·17-s + 0.648·19-s + 0.834·23-s + 1.08·27-s + 1.48·29-s + 1.96·33-s + 1.31·37-s − 1.92·39-s − 1.10·41-s + 0.609·43-s − 0.825·47-s − 0.560·51-s − 1.37·53-s + 1.05·57-s + 1.84·59-s − 0.905·61-s + 1.36·69-s + 0.827·73-s + 0.900·79-s + 0.111·81-s + 1.55·83-s + 2.42·87-s + 0.749·89-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4900$$    =    $$2^{2} \cdot 5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$39.1266$$ Root analytic conductor: $$6.25513$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4900,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.995993141$$ $$L(\frac12)$$ $$\approx$$ $$3.995993141$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1$$
good3 $$1 - 2.82T + 3T^{2}$$
11 $$1 - 4T + 11T^{2}$$
13 $$1 + 4.24T + 13T^{2}$$
17 $$1 + 1.41T + 17T^{2}$$
19 $$1 - 2.82T + 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 - 8T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 8T + 37T^{2}$$
41 $$1 + 7.07T + 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 + 5.65T + 47T^{2}$$
53 $$1 + 10T + 53T^{2}$$
59 $$1 - 14.1T + 59T^{2}$$
61 $$1 + 7.07T + 61T^{2}$$
67 $$1 + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 7.07T + 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 14.1T + 83T^{2}$$
89 $$1 - 7.07T + 89T^{2}$$
97 $$1 - 1.41T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$