# Properties

 Label 2-9800-1.1-c1-0-140 Degree $2$ Conductor $9800$ Sign $-1$ Analytic cond. $78.2533$ Root an. cond. $8.84609$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.414·3-s − 2.82·9-s + 0.828·11-s − 2·13-s + 7.65·17-s − 5.65·19-s + 5.58·23-s + 2.41·27-s − 7.82·29-s + 0.828·31-s − 0.343·33-s − 5.65·37-s + 0.828·39-s + 5.82·41-s + 6.89·43-s − 11.6·47-s − 3.17·51-s + 5.65·53-s + 2.34·57-s − 4·59-s + 6.65·61-s + 12.8·67-s − 2.31·69-s − 12·71-s − 3.65·73-s − 4·79-s + 7.48·81-s + ⋯
 L(s)  = 1 − 0.239·3-s − 0.942·9-s + 0.249·11-s − 0.554·13-s + 1.85·17-s − 1.29·19-s + 1.16·23-s + 0.464·27-s − 1.45·29-s + 0.148·31-s − 0.0597·33-s − 0.929·37-s + 0.132·39-s + 0.910·41-s + 1.05·43-s − 1.70·47-s − 0.444·51-s + 0.777·53-s + 0.310·57-s − 0.520·59-s + 0.852·61-s + 1.57·67-s − 0.278·69-s − 1.42·71-s − 0.428·73-s − 0.450·79-s + 0.831·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9800$$    =    $$2^{3} \cdot 5^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$78.2533$$ Root analytic conductor: $$8.84609$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{9800} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9800,\ (\ :1/2),\ -1)$$

## 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 + 0.414T + 3T^{2}$$
11 $$1 - 0.828T + 11T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 - 7.65T + 17T^{2}$$
19 $$1 + 5.65T + 19T^{2}$$
23 $$1 - 5.58T + 23T^{2}$$
29 $$1 + 7.82T + 29T^{2}$$
31 $$1 - 0.828T + 31T^{2}$$
37 $$1 + 5.65T + 37T^{2}$$
41 $$1 - 5.82T + 41T^{2}$$
43 $$1 - 6.89T + 43T^{2}$$
47 $$1 + 11.6T + 47T^{2}$$
53 $$1 - 5.65T + 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 - 6.65T + 61T^{2}$$
67 $$1 - 12.8T + 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + 3.65T + 73T^{2}$$
79 $$1 + 4T + 79T^{2}$$
83 $$1 - 4.75T + 83T^{2}$$
89 $$1 + 5.34T + 89T^{2}$$
97 $$1 + 6T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$