# Properties

 Label 2-11e2-1.1-c3-0-9 Degree $2$ Conductor $121$ Sign $1$ Analytic cond. $7.13923$ Root an. cond. $2.67193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 8·3-s − 8·4-s + 18·5-s + 37·9-s − 64·12-s + 144·15-s + 64·16-s − 144·20-s − 108·23-s + 199·25-s + 80·27-s + 340·31-s − 296·36-s − 434·37-s + 666·45-s − 36·47-s + 512·48-s − 343·49-s − 738·53-s − 720·59-s − 1.15e3·60-s − 512·64-s − 416·67-s − 864·69-s + 612·71-s + 1.59e3·75-s + 1.15e3·80-s + ⋯
 L(s)  = 1 + 1.53·3-s − 4-s + 1.60·5-s + 1.37·9-s − 1.53·12-s + 2.47·15-s + 16-s − 1.60·20-s − 0.979·23-s + 1.59·25-s + 0.570·27-s + 1.96·31-s − 1.37·36-s − 1.92·37-s + 2.20·45-s − 0.111·47-s + 1.53·48-s − 49-s − 1.91·53-s − 1.58·59-s − 2.47·60-s − 64-s − 0.758·67-s − 1.50·69-s + 1.02·71-s + 2.45·75-s + 1.60·80-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$121$$    =    $$11^{2}$$ Sign: $1$ Analytic conductor: $$7.13923$$ Root analytic conductor: $$2.67193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 121,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.657507929$$ $$L(\frac12)$$ $$\approx$$ $$2.657507929$$ $$L(\frac{5}{2})$$ 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^{3} T^{2}$$
3 $$1 - 8 T + p^{3} T^{2}$$
5 $$1 - 18 T + p^{3} T^{2}$$
7 $$1 + p^{3} T^{2}$$
13 $$1 + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + p^{3} T^{2}$$
23 $$1 + 108 T + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 - 340 T + p^{3} T^{2}$$
37 $$1 + 434 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + p^{3} T^{2}$$
47 $$1 + 36 T + p^{3} T^{2}$$
53 $$1 + 738 T + p^{3} T^{2}$$
59 $$1 + 720 T + p^{3} T^{2}$$
61 $$1 + p^{3} T^{2}$$
67 $$1 + 416 T + p^{3} T^{2}$$
71 $$1 - 612 T + p^{3} T^{2}$$
73 $$1 + p^{3} T^{2}$$
79 $$1 + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 - 1674 T + p^{3} T^{2}$$
97 $$1 + 34 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$