# Properties

 Degree 2 Conductor $3 \cdot 5^{2} \cdot 7$ Sign $-0.691 - 0.722i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.448 + 1.67i)3-s + (−1.73 − i)4-s + (1.48 + 2.19i)7-s + (−2.59 + 1.50i)9-s + (0.896 − 3.34i)12-s + (−1.22 + 1.22i)13-s + (1.99 + 3.46i)16-s + (−6.06 + 3.5i)19-s + (−2.99 + 3.46i)21-s + (−3.67 − 3.67i)27-s + (−0.378 − 5.27i)28-s + (−3.5 + 6.06i)31-s + 6·36-s + (−3.13 + 11.7i)37-s + (−2.59 − 1.5i)39-s + ⋯
 L(s)  = 1 + (0.258 + 0.965i)3-s + (−0.866 − 0.5i)4-s + (0.560 + 0.827i)7-s + (−0.866 + 0.5i)9-s + (0.258 − 0.965i)12-s + (−0.339 + 0.339i)13-s + (0.499 + 0.866i)16-s + (−1.39 + 0.802i)19-s + (−0.654 + 0.755i)21-s + (−0.707 − 0.707i)27-s + (−0.0716 − 0.997i)28-s + (−0.628 + 1.08i)31-s + 36-s + (−0.515 + 1.92i)37-s + (−0.416 − 0.240i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$525$$    =    $$3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.691 - 0.722i$ motivic weight = $$1$$ character : $\chi_{525} (32, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 525,\ (\ :1/2),\ -0.691 - 0.722i)$$ $$L(1)$$ $$\approx$$ $$0.369305 + 0.864945i$$ $$L(\frac12)$$ $$\approx$$ $$0.369305 + 0.864945i$$ $$L(\frac{3}{2})$$ 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 + (-0.448 - 1.67i)T$$
5 $$1$$
7 $$1 + (-1.48 - 2.19i)T$$
good2 $$1 + (1.73 + i)T^{2}$$
11 $$1 + (5.5 + 9.52i)T^{2}$$
13 $$1 + (1.22 - 1.22i)T - 13iT^{2}$$
17 $$1 + (-14.7 + 8.5i)T^{2}$$
19 $$1 + (6.06 - 3.5i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-19.9 - 11.5i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (3.13 - 11.7i)T + (-32.0 - 18.5i)T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + (-8.57 + 8.57i)T - 43iT^{2}$$
47 $$1 + (40.7 + 23.5i)T^{2}$$
53 $$1 + (45.8 - 26.5i)T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-11.7 + 3.13i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (-4.03 - 15.0i)T + (-63.2 + 36.5i)T^{2}$$
79 $$1 + (-11.2 + 6.5i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-9.79 - 9.79i)T + 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}