# Properties

 Label 2-70e2-1.1-c1-0-63 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.56·3-s + 3.59·9-s − 1.56·11-s − 5.56·13-s − 7.16·17-s + 3.16·19-s − 5.73·23-s + 1.53·27-s + 1.96·29-s + 0.969·31-s − 4.03·33-s + 6.70·37-s − 14.3·39-s − 8.87·41-s + 4.59·43-s − 0.401·47-s − 18.4·51-s − 9.53·53-s + 8.13·57-s − 4.56·59-s + 15.3·61-s − 0.862·67-s − 14.7·69-s − 12.1·71-s − 4·73-s − 12.8·79-s − 6.84·81-s + ⋯
 L(s)  = 1 + 1.48·3-s + 1.19·9-s − 0.473·11-s − 1.54·13-s − 1.73·17-s + 0.726·19-s − 1.19·23-s + 0.296·27-s + 0.365·29-s + 0.174·31-s − 0.701·33-s + 1.10·37-s − 2.29·39-s − 1.38·41-s + 0.701·43-s − 0.0584·47-s − 2.57·51-s − 1.31·53-s + 1.07·57-s − 0.594·59-s + 1.95·61-s − 0.105·67-s − 1.77·69-s − 1.44·71-s − 0.468·73-s − 1.44·79-s − 0.760·81-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: $$1$$ Selberg data: $$(2,\ 4900,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 - 2.56T + 3T^{2}$$
11 $$1 + 1.56T + 11T^{2}$$
13 $$1 + 5.56T + 13T^{2}$$
17 $$1 + 7.16T + 17T^{2}$$
19 $$1 - 3.16T + 19T^{2}$$
23 $$1 + 5.73T + 23T^{2}$$
29 $$1 - 1.96T + 29T^{2}$$
31 $$1 - 0.969T + 31T^{2}$$
37 $$1 - 6.70T + 37T^{2}$$
41 $$1 + 8.87T + 41T^{2}$$
43 $$1 - 4.59T + 43T^{2}$$
47 $$1 + 0.401T + 47T^{2}$$
53 $$1 + 9.53T + 53T^{2}$$
59 $$1 + 4.56T + 59T^{2}$$
61 $$1 - 15.3T + 61T^{2}$$
67 $$1 + 0.862T + 67T^{2}$$
71 $$1 + 12.1T + 71T^{2}$$
73 $$1 + 4T + 73T^{2}$$
79 $$1 + 12.8T + 79T^{2}$$
83 $$1 + 17.1T + 83T^{2}$$
89 $$1 + 5.59T + 89T^{2}$$
97 $$1 + 0.233T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$