# Properties

 Degree 2 Conductor $2 \cdot 3^{3} \cdot 5^{2} \cdot 7$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + 7-s + 8-s + 4·11-s + 6·13-s + 14-s + 16-s + 4·17-s + 4·22-s + 4·23-s + 6·26-s + 28-s − 3·29-s + 7·31-s + 32-s + 4·34-s − 37-s + 7·41-s − 10·43-s + 4·44-s + 4·46-s − 13·47-s + 49-s + 6·52-s + 6·53-s + 56-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.852·22-s + 0.834·23-s + 1.17·26-s + 0.188·28-s − 0.557·29-s + 1.25·31-s + 0.176·32-s + 0.685·34-s − 0.164·37-s + 1.09·41-s − 1.52·43-s + 0.603·44-s + 0.589·46-s − 1.89·47-s + 1/7·49-s + 0.832·52-s + 0.824·53-s + 0.133·56-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$9450$$    =    $$2 \cdot 3^{3} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{9450} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 9450,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $4.818615920$ $L(\frac12)$ $\approx$ $4.818615920$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;7\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - T$$
3 $$1$$
5 $$1$$
7 $$1 - T$$
good11 $$1 - 4 T + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 + 3 T + p T^{2}$$
31 $$1 - 7 T + p T^{2}$$
37 $$1 + T + p T^{2}$$
41 $$1 - 7 T + p T^{2}$$
43 $$1 + 10 T + p T^{2}$$
47 $$1 + 13 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 5 T + p T^{2}$$
61 $$1 + 7 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 3 T + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 - 12 T + p T^{2}$$
83 $$1 + 2 T + p T^{2}$$
89 $$1 + 14 T + p T^{2}$$
97 $$1 - 10 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}