# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·13-s + 6·17-s + 4·19-s + 6·23-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s + 10·43-s + 6·47-s − 6·53-s + 12·59-s − 2·61-s − 2·67-s + 12·71-s + 2·73-s + 8·79-s − 6·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
 L(s)  = 1 + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.244·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 0.658·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$44100$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{44100} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 44100,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $3.307569488$ $L(\frac12)$ $\approx$ $3.307569488$ $L(\frac{3}{2})$ 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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
7 $$1$$
good11 $$1 + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 10 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 2 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}