# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $-0.904 + 0.426i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.62 + 0.358i)7-s + 0.999i·8-s + (2.59 − 1.5i)11-s + 2.44i·13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (−2.95 − 5.12i)17-s + (−5.12 − 2.95i)19-s + 3·22-s + (3.67 + 2.12i)23-s + (−1.22 + 2.12i)26-s + (−1.62 − 2.09i)28-s + 7.24i·29-s + ⋯
 L(s)  = 1 + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.990 + 0.135i)7-s + 0.353i·8-s + (0.783 − 0.452i)11-s + 0.679i·13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.717 − 1.24i)17-s + (−1.17 − 0.678i)19-s + 0.639·22-s + (0.766 + 0.442i)23-s + (−0.240 + 0.416i)26-s + (−0.306 − 0.395i)28-s + 1.34i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.904 + 0.426i$ motivic weight = $$1$$ character : $\chi_{3150} (1601, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ -0.904 + 0.426i)$ $L(1)$ $\approx$ $0.2222066651$ $L(\frac12)$ $\approx$ $0.2222066651$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.866 - 0.5i)T$$
3 $$1$$
5 $$1$$
7 $$1 + (2.62 - 0.358i)T$$
good11 $$1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 - 2.44iT - 13T^{2}$$
17 $$1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 - 7.24iT - 29T^{2}$$
31 $$1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 11.8T + 41T^{2}$$
43 $$1 - 0.242T + 43T^{2}$$
47 $$1 + (2.95 - 5.12i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (6.27 - 3.62i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (4.03 + 6.98i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 1.75iT - 71T^{2}$$
73 $$1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 6.63T + 83T^{2}$$
89 $$1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 13.5iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}