# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 9·3-s − 94·7-s + 81·9-s + 146·13-s − 46·19-s − 846·21-s + 625·25-s + 729·27-s + 194·31-s − 2.06e3·37-s + 1.31e3·39-s − 3.21e3·43-s + 6.43e3·49-s − 414·57-s − 1.96e3·61-s − 7.61e3·63-s + 5.90e3·67-s − 8.54e3·73-s + 5.62e3·75-s + 7.68e3·79-s + 6.56e3·81-s − 1.37e4·91-s + 1.74e3·93-s − 1.88e4·97-s + 1.64e4·103-s + 2.20e4·109-s − 1.85e4·111-s + ⋯
 L(s)  = 1 + 3-s − 1.91·7-s + 9-s + 0.863·13-s − 0.127·19-s − 1.91·21-s + 25-s + 27-s + 0.201·31-s − 1.50·37-s + 0.863·39-s − 1.73·43-s + 2.68·49-s − 0.127·57-s − 0.528·61-s − 1.91·63-s + 1.31·67-s − 1.60·73-s + 75-s + 1.23·79-s + 81-s − 1.65·91-s + 0.201·93-s − 1.99·97-s + 1.54·103-s + 1.85·109-s − 1.50·111-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : $\chi_{12} (5, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :2),\ 1)$ $L(\frac{5}{2})$ $\approx$ $1.27025$ $L(\frac12)$ $\approx$ $1.27025$ $L(3)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - p^{2} T$$
good5 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
7 $$1 + 94 T + p^{4} T^{2}$$
11 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
13 $$1 - 146 T + p^{4} T^{2}$$
17 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
19 $$1 + 46 T + p^{4} T^{2}$$
23 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
29 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
31 $$1 - 194 T + p^{4} T^{2}$$
37 $$1 + 2062 T + p^{4} T^{2}$$
41 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
43 $$1 + 3214 T + p^{4} T^{2}$$
47 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
53 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
59 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
61 $$1 + 1966 T + p^{4} T^{2}$$
67 $$1 - 5906 T + p^{4} T^{2}$$
71 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
73 $$1 + 8542 T + p^{4} T^{2}$$
79 $$1 - 7682 T + p^{4} T^{2}$$
83 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
89 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
97 $$1 + 18814 T + p^{4} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}