# Properties

 Degree 2 Conductor $5^{2} \cdot 7^{2}$ Sign $0.850 + 0.525i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − i·4-s + i·9-s + 2·11-s − 16-s − 2i·29-s + 36-s − 2i·44-s + i·64-s − 2·71-s + 2i·79-s − 81-s + 2i·99-s + 2i·109-s − 2·116-s + ⋯
 L(s)  = 1 − i·4-s + i·9-s + 2·11-s − 16-s − 2i·29-s + 36-s − 2i·44-s + i·64-s − 2·71-s + 2i·79-s − 81-s + 2i·99-s + 2i·109-s − 2·116-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1225$$    =    $$5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $0.850 + 0.525i$ motivic weight = $$0$$ character : $\chi_{1225} (932, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 1225,\ (\ :0),\ 0.850 + 0.525i)$ $L(\frac{1}{2})$ $\approx$ $1.152995143$ $L(\frac12)$ $\approx$ $1.152995143$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;7\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1$$
7 $$1$$
good2 $$1 + iT^{2}$$
3 $$1 - iT^{2}$$
11 $$1 - 2T + T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 + 2iT - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 - iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + iT^{2}$$
71 $$1 + 2T + T^{2}$$
73 $$1 - iT^{2}$$
79 $$1 - 2iT - T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 - T^{2}$$
97 $$1 + iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}