# Properties

 Degree 2 Conductor $3 \cdot 5^{2} \cdot 7$ Sign $0.792 + 0.609i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.800 − 0.800i)2-s + (1.34 − 1.09i)3-s + 0.718i·4-s + (0.199 − 1.95i)6-s + (0.707 + 0.707i)7-s + (2.17 + 2.17i)8-s + (0.606 − 2.93i)9-s + 5.20i·11-s + (0.785 + 0.964i)12-s + (3.24 − 3.24i)13-s + 1.13·14-s + 2.04·16-s + (0.844 − 0.844i)17-s + (−1.86 − 2.83i)18-s − 1.32i·19-s + ⋯
 L(s)  = 1 + (0.566 − 0.566i)2-s + (0.775 − 0.631i)3-s + 0.359i·4-s + (0.0813 − 0.796i)6-s + (0.267 + 0.267i)7-s + (0.769 + 0.769i)8-s + (0.202 − 0.979i)9-s + 1.56i·11-s + (0.226 + 0.278i)12-s + (0.900 − 0.900i)13-s + 0.302·14-s + 0.511·16-s + (0.204 − 0.204i)17-s + (−0.439 − 0.668i)18-s − 0.302i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$525$$    =    $$3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.792 + 0.609i$ motivic weight = $$1$$ character : $\chi_{525} (407, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 525,\ (\ :1/2),\ 0.792 + 0.609i)$$ $$L(1)$$ $$\approx$$ $$2.47533 - 0.841445i$$ $$L(\frac12)$$ $$\approx$$ $$2.47533 - 0.841445i$$ $$L(\frac{3}{2})$$ 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 + (-1.34 + 1.09i)T$$
5 $$1$$
7 $$1 + (-0.707 - 0.707i)T$$
good2 $$1 + (-0.800 + 0.800i)T - 2iT^{2}$$
11 $$1 - 5.20iT - 11T^{2}$$
13 $$1 + (-3.24 + 3.24i)T - 13iT^{2}$$
17 $$1 + (-0.844 + 0.844i)T - 17iT^{2}$$
19 $$1 + 1.32iT - 19T^{2}$$
23 $$1 + (5.62 + 5.62i)T + 23iT^{2}$$
29 $$1 + 4.38T + 29T^{2}$$
31 $$1 + 1.70T + 31T^{2}$$
37 $$1 + (-1.71 - 1.71i)T + 37iT^{2}$$
41 $$1 - 1.82iT - 41T^{2}$$
43 $$1 + (-0.281 + 0.281i)T - 43iT^{2}$$
47 $$1 + (3.39 - 3.39i)T - 47iT^{2}$$
53 $$1 + (-3.51 - 3.51i)T + 53iT^{2}$$
59 $$1 + 1.81T + 59T^{2}$$
61 $$1 + 2.47T + 61T^{2}$$
67 $$1 + (7.92 + 7.92i)T + 67iT^{2}$$
71 $$1 + 9.06iT - 71T^{2}$$
73 $$1 + (-1.33 + 1.33i)T - 73iT^{2}$$
79 $$1 - 11.5iT - 79T^{2}$$
83 $$1 + (5.46 + 5.46i)T + 83iT^{2}$$
89 $$1 - 9.43T + 89T^{2}$$
97 $$1 + (-3.06 - 3.06i)T + 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}