# Properties

 Label 2-9800-1.1-c1-0-43 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 $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.18·3-s − 1.59·9-s + 0.230·11-s − 2.27·13-s + 6.53·17-s + 0.260·19-s + 8.87·23-s + 5.44·27-s + 5.42·29-s − 4.87·31-s − 0.272·33-s + 1.15·37-s + 2.69·39-s − 4.43·41-s − 4.17·43-s + 0.923·47-s − 7.73·51-s + 2.13·53-s − 0.308·57-s + 5.07·59-s − 8.88·61-s − 8.15·67-s − 10.5·69-s − 9.06·71-s + 6.00·73-s + 0.112·79-s − 1.65·81-s + ⋯
 L(s)  = 1 − 0.683·3-s − 0.532·9-s + 0.0694·11-s − 0.630·13-s + 1.58·17-s + 0.0596·19-s + 1.84·23-s + 1.04·27-s + 1.00·29-s − 0.874·31-s − 0.0474·33-s + 0.189·37-s + 0.430·39-s − 0.692·41-s − 0.637·43-s + 0.134·47-s − 1.08·51-s + 0.293·53-s − 0.0408·57-s + 0.660·59-s − 1.13·61-s − 0.996·67-s − 1.26·69-s − 1.07·71-s + 0.703·73-s + 0.0127·79-s − 0.183·81-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: $$0$$ Selberg data: $$(2,\ 9800,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.396396344$$ $$L(\frac12)$$ $$\approx$$ $$1.396396344$$ $$L(\frac{3}{2})$$ 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 + 1.18T + 3T^{2}$$
11 $$1 - 0.230T + 11T^{2}$$
13 $$1 + 2.27T + 13T^{2}$$
17 $$1 - 6.53T + 17T^{2}$$
19 $$1 - 0.260T + 19T^{2}$$
23 $$1 - 8.87T + 23T^{2}$$
29 $$1 - 5.42T + 29T^{2}$$
31 $$1 + 4.87T + 31T^{2}$$
37 $$1 - 1.15T + 37T^{2}$$
41 $$1 + 4.43T + 41T^{2}$$
43 $$1 + 4.17T + 43T^{2}$$
47 $$1 - 0.923T + 47T^{2}$$
53 $$1 - 2.13T + 53T^{2}$$
59 $$1 - 5.07T + 59T^{2}$$
61 $$1 + 8.88T + 61T^{2}$$
67 $$1 + 8.15T + 67T^{2}$$
71 $$1 + 9.06T + 71T^{2}$$
73 $$1 - 6.00T + 73T^{2}$$
79 $$1 - 0.112T + 79T^{2}$$
83 $$1 - 8.72T + 83T^{2}$$
89 $$1 + 15.9T + 89T^{2}$$
97 $$1 - 4.29T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$