# Properties

 Degree 2 Conductor $11^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s − 2·9-s + 2·10-s − 2·12-s − 4·13-s + 4·14-s − 15-s − 4·16-s + 2·17-s − 4·18-s + 2·20-s − 2·21-s − 23-s − 4·25-s − 8·26-s + 5·27-s + 4·28-s − 2·30-s + 7·31-s − 8·32-s + 4·34-s + 2·35-s + ⋯
 L(s)  = 1 + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s − 16-s + 0.485·17-s − 0.942·18-s + 0.447·20-s − 0.436·21-s − 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s + 0.755·28-s − 0.365·30-s + 1.25·31-s − 1.41·32-s + 0.685·34-s + 0.338·35-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

 $$d$$ = $$2$$ $$N$$ = $$121$$    =    $$11^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{121} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 121,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.75939$ $L(\frac12)$ $\approx$ $1.75939$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 $$1$$
good2 $$1 - p T + p T^{2}$$
3 $$1 + T + p T^{2}$$
5 $$1 - T + p T^{2}$$
7 $$1 - 2 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 - 7 T + p T^{2}$$
37 $$1 - 3 T + p T^{2}$$
41 $$1 - 8 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 5 T + p T^{2}$$
61 $$1 + 12 T + p T^{2}$$
67 $$1 + 7 T + p T^{2}$$
71 $$1 + 3 T + p T^{2}$$
73 $$1 + 4 T + p T^{2}$$
79 $$1 - 10 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 - 15 T + p T^{2}$$
97 $$1 + 7 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}