# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·5-s − 2·13-s − 4·17-s − 6·19-s − 25-s − 8·29-s + 8·31-s + 10·37-s − 8·41-s + 2·43-s − 8·47-s + 2·53-s + 12·59-s + 10·61-s − 4·65-s + 12·67-s − 8·71-s + 6·73-s + 2·79-s − 16·83-s − 8·85-s − 14·89-s − 12·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
 L(s)  = 1 + 0.894·5-s − 0.554·13-s − 0.970·17-s − 1.37·19-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s + 0.274·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.225·79-s − 1.75·83-s − 0.867·85-s − 1.48·89-s − 1.23·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$213444$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{213444} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 213444,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7,\;11\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1$$
11 $$1$$
good5 $$1 - 2 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 8 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 + 8 T + p T^{2}$$
43 $$1 - 2 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 2 T + p T^{2}$$
83 $$1 + 16 T + p T^{2}$$
89 $$1 + 14 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}