# Properties

 Label 2-9702-1.1-c1-0-98 Degree $2$ Conductor $9702$ Sign $-1$ Analytic cond. $77.4708$ Root an. cond. $8.80175$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 11-s + 4·13-s + 16-s − 6·17-s + 2·19-s − 2·20-s + 22-s − 25-s − 4·26-s + 6·29-s + 2·31-s − 32-s + 6·34-s + 2·37-s − 2·38-s + 2·40-s + 2·41-s − 12·43-s − 44-s + 6·47-s + 50-s + 4·52-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s − 0.784·26-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.316·40-s + 0.312·41-s − 1.82·43-s − 0.150·44-s + 0.875·47-s + 0.141·50-s + 0.554·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9702$$    =    $$2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$77.4708$$ Root analytic conductor: $$8.80175$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9702,\ (\ :1/2),\ -1)$$

## 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 + T$$
3 $$1$$
7 $$1$$
11 $$1 + T$$
good5 $$1 + 2 T + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 2 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 + 8 T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 12 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 - 12 T + p T^{2}$$
97 $$1 + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$