# Properties

 Label 2-70e2-1.1-c1-0-23 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 + 1.73·3-s + 2.27·11-s − 6.09·13-s + 4.77·17-s + 4.27·19-s − 0.894·23-s − 5.19·27-s − 3.27·29-s + 4.27·31-s + 3.94·33-s + 5.61·37-s − 10.5·39-s + 11.2·41-s + 6.50·43-s + 2.15·47-s + 8.27·51-s − 7.40·53-s + 7.40·57-s − 4.27·59-s − 1.54·61-s + 13.9·67-s − 1.54·69-s + 10.5·71-s + 2.15·73-s − 0.274·79-s − 9·81-s + 5.67·83-s + ⋯
 L(s)  = 1 + 1.00·3-s + 0.685·11-s − 1.68·13-s + 1.15·17-s + 0.980·19-s − 0.186·23-s − 1.00·27-s − 0.608·29-s + 0.767·31-s + 0.685·33-s + 0.923·37-s − 1.68·39-s + 1.76·41-s + 0.992·43-s + 0.313·47-s + 1.15·51-s − 1.01·53-s + 0.980·57-s − 0.556·59-s − 0.198·61-s + 1.69·67-s − 0.186·69-s + 1.25·71-s + 0.251·73-s − 0.0309·79-s − 81-s + 0.622·83-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: $\chi_{4900} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4900,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.764705749$$ $$L(\frac12)$$ $$\approx$$ $$2.764705749$$ $$L(\frac{3}{2})$$ 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 - 1.73T + 3T^{2}$$
11 $$1 - 2.27T + 11T^{2}$$
13 $$1 + 6.09T + 13T^{2}$$
17 $$1 - 4.77T + 17T^{2}$$
19 $$1 - 4.27T + 19T^{2}$$
23 $$1 + 0.894T + 23T^{2}$$
29 $$1 + 3.27T + 29T^{2}$$
31 $$1 - 4.27T + 31T^{2}$$
37 $$1 - 5.61T + 37T^{2}$$
41 $$1 - 11.2T + 41T^{2}$$
43 $$1 - 6.50T + 43T^{2}$$
47 $$1 - 2.15T + 47T^{2}$$
53 $$1 + 7.40T + 53T^{2}$$
59 $$1 + 4.27T + 59T^{2}$$
61 $$1 + 1.54T + 61T^{2}$$
67 $$1 - 13.9T + 67T^{2}$$
71 $$1 - 10.5T + 71T^{2}$$
73 $$1 - 2.15T + 73T^{2}$$
79 $$1 + 0.274T + 79T^{2}$$
83 $$1 - 5.67T + 83T^{2}$$
89 $$1 - 7T + 89T^{2}$$
97 $$1 + 6.92T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$