# Properties

 Degree 2 Conductor $3 \cdot 5^{2} \cdot 7$ Sign $0.573 - 0.818i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.48 − 1.48i)2-s + (0.707 + 0.707i)3-s + 2.43i·4-s − 2.10i·6-s + (1.75 + 1.97i)7-s + (0.640 − 0.640i)8-s + 1.00i·9-s − 2.67·11-s + (−1.71 + 1.71i)12-s + (1.22 + 1.22i)13-s + (0.320 − 5.55i)14-s + 2.95·16-s + (−4.74 + 4.74i)17-s + (1.48 − 1.48i)18-s − 6.01·19-s + ⋯
 L(s)  = 1 + (−1.05 − 1.05i)2-s + (0.408 + 0.408i)3-s + 1.21i·4-s − 0.859i·6-s + (0.665 + 0.746i)7-s + (0.226 − 0.226i)8-s + 0.333i·9-s − 0.805·11-s + (−0.496 + 0.496i)12-s + (0.340 + 0.340i)13-s + (0.0857 − 1.48i)14-s + 0.738·16-s + (−1.15 + 1.15i)17-s + (0.350 − 0.350i)18-s − 1.38·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$525$$    =    $$3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.573 - 0.818i$ motivic weight = $$1$$ character : $\chi_{525} (307, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 525,\ (\ :1/2),\ 0.573 - 0.818i)$$ $$L(1)$$ $$\approx$$ $$0.610323 + 0.317574i$$ $$L(\frac12)$$ $$\approx$$ $$0.610323 + 0.317574i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-0.707 - 0.707i)T$$
5 $$1$$
7 $$1 + (-1.75 - 1.97i)T$$
good2 $$1 + (1.48 + 1.48i)T + 2iT^{2}$$
11 $$1 + 2.67T + 11T^{2}$$
13 $$1 + (-1.22 - 1.22i)T + 13iT^{2}$$
17 $$1 + (4.74 - 4.74i)T - 17iT^{2}$$
19 $$1 + 6.01T + 19T^{2}$$
23 $$1 + (-0.175 + 0.175i)T - 23iT^{2}$$
29 $$1 + 0.304iT - 29T^{2}$$
31 $$1 - 7.25iT - 31T^{2}$$
37 $$1 + (-0.735 - 0.735i)T + 37iT^{2}$$
41 $$1 - 7.05iT - 41T^{2}$$
43 $$1 + (0.304 - 0.304i)T - 43iT^{2}$$
47 $$1 + (-0.556 + 0.556i)T - 47iT^{2}$$
53 $$1 + (-4.99 + 4.99i)T - 53iT^{2}$$
59 $$1 - 7.98T + 59T^{2}$$
61 $$1 + 5.53iT - 61T^{2}$$
67 $$1 + (-3.43 - 3.43i)T + 67iT^{2}$$
71 $$1 - 15.3T + 71T^{2}$$
73 $$1 + (-10.0 - 10.0i)T + 73iT^{2}$$
79 $$1 + 11.2iT - 79T^{2}$$
83 $$1 + (4.88 + 4.88i)T + 83iT^{2}$$
89 $$1 + 6.91T + 89T^{2}$$
97 $$1 + (8.84 - 8.84i)T - 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}