# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2}$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯
 L(s)  = 1 − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$144$$    =    $$2^{4} \cdot 3^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{144} (127, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 144,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.6120575525$ $L(\frac12)$ $\approx$ $0.6120575525$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
good5 $$1 + T^{2}$$
7 $$( 1 - T )( 1 + T )$$
11 $$( 1 - T )( 1 + T )$$
13 $$( 1 + T )^{2}$$
17 $$1 + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$( 1 - T )( 1 + T )$$
29 $$1 + T^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )^{2}$$
41 $$1 + T^{2}$$
43 $$( 1 - T )( 1 + T )$$
47 $$( 1 - T )( 1 + T )$$
53 $$1 + T^{2}$$
59 $$( 1 - T )( 1 + T )$$
61 $$( 1 - T )^{2}$$
67 $$( 1 - T )( 1 + T )$$
71 $$( 1 - T )( 1 + T )$$
73 $$( 1 + T )^{2}$$
79 $$( 1 - T )( 1 + T )$$
83 $$( 1 - T )( 1 + T )$$
89 $$1 + T^{2}$$
97 $$( 1 + T )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}