# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 25·5-s − 62·7-s + 144·11-s − 654·13-s + 1.19e3·17-s + 556·19-s − 2.18e3·23-s + 625·25-s + 1.57e3·29-s + 9.66e3·31-s − 1.55e3·35-s − 3.53e3·37-s − 7.46e3·41-s − 7.11e3·43-s + 2.82e4·47-s − 1.29e4·49-s + 1.30e4·53-s + 3.60e3·55-s + 3.70e4·59-s + 3.95e4·61-s − 1.63e4·65-s − 5.67e4·67-s − 4.55e4·71-s + 1.18e4·73-s − 8.92e3·77-s + 9.42e4·79-s + 3.14e4·83-s + ⋯
 L(s)  = 1 + 0.447·5-s − 0.478·7-s + 0.358·11-s − 1.07·13-s + 0.998·17-s + 0.353·19-s − 0.860·23-s + 1/5·25-s + 0.348·29-s + 1.80·31-s − 0.213·35-s − 0.424·37-s − 0.693·41-s − 0.586·43-s + 1.86·47-s − 0.771·49-s + 0.637·53-s + 0.160·55-s + 1.38·59-s + 1.36·61-s − 0.479·65-s − 1.54·67-s − 1.07·71-s + 0.260·73-s − 0.171·77-s + 1.69·79-s + 0.501·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$360$$    =    $$2^{3} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{360} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 360,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$2.074348642$$ $$L(\frac12)$$ $$\approx$$ $$2.074348642$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 - p^{2} T$$
good7 $$1 + 62 T + p^{5} T^{2}$$
11 $$1 - 144 T + p^{5} T^{2}$$
13 $$1 + 654 T + p^{5} T^{2}$$
17 $$1 - 70 p T + p^{5} T^{2}$$
19 $$1 - 556 T + p^{5} T^{2}$$
23 $$1 + 2182 T + p^{5} T^{2}$$
29 $$1 - 1578 T + p^{5} T^{2}$$
31 $$1 - 9660 T + p^{5} T^{2}$$
37 $$1 + 3534 T + p^{5} T^{2}$$
41 $$1 + 182 p T + p^{5} T^{2}$$
43 $$1 + 7114 T + p^{5} T^{2}$$
47 $$1 - 602 p T + p^{5} T^{2}$$
53 $$1 - 13046 T + p^{5} T^{2}$$
59 $$1 - 37092 T + p^{5} T^{2}$$
61 $$1 - 39570 T + p^{5} T^{2}$$
67 $$1 + 56734 T + p^{5} T^{2}$$
71 $$1 + 45588 T + p^{5} T^{2}$$
73 $$1 - 11842 T + p^{5} T^{2}$$
79 $$1 - 94216 T + p^{5} T^{2}$$
83 $$1 - 31482 T + p^{5} T^{2}$$
89 $$1 - 94054 T + p^{5} T^{2}$$
97 $$1 - 23714 T + p^{5} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}