# Properties

 Degree $2$ Conductor $121$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 2·4-s − 3·5-s − 2·9-s + 2·12-s + 3·15-s + 4·16-s + 6·20-s − 9·23-s + 4·25-s + 5·27-s − 5·31-s + 4·36-s + 7·37-s + 6·45-s − 12·47-s − 4·48-s − 7·49-s + 6·53-s − 15·59-s − 6·60-s − 8·64-s + 13·67-s + 9·69-s − 3·71-s − 4·75-s − 12·80-s + ⋯
 L(s)  = 1 − 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.577·12-s + 0.774·15-s + 16-s + 1.34·20-s − 1.87·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 2/3·36-s + 1.15·37-s + 0.894·45-s − 1.75·47-s − 0.577·48-s − 49-s + 0.824·53-s − 1.95·59-s − 0.774·60-s − 64-s + 1.58·67-s + 1.08·69-s − 0.356·71-s − 0.461·75-s − 1.34·80-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$121$$    =    $$11^{2}$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{121} (1, \cdot )$ Sato-Tate group: $N(\mathrm{U}(1))$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 121,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad11 $$1$$
good2 $$1 + p T^{2}$$
3 $$1 + T + p T^{2}$$
5 $$1 + 3 T + p T^{2}$$
7 $$1 + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + 9 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 5 T + p T^{2}$$
37 $$1 - 7 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 15 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 13 T + p T^{2}$$
71 $$1 + 3 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + 9 T + p T^{2}$$
97 $$1 - 17 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$