# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $0.349 + 0.936i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.80·2-s − 1.73i·3-s + 10.4·4-s − 2.23i·5-s + 6.58i·6-s + (2.44 + 6.55i)7-s − 24.6·8-s − 2.99·9-s + 8.50i·10-s + 14.4·11-s − 18.1i·12-s − 16.9i·13-s + (−9.30 − 24.9i)14-s − 3.87·15-s + 51.8·16-s − 13.0i·17-s + ⋯
 L(s)  = 1 − 1.90·2-s − 0.577i·3-s + 2.61·4-s − 0.447i·5-s + 1.09i·6-s + (0.349 + 0.936i)7-s − 3.07·8-s − 0.333·9-s + 0.850i·10-s + 1.31·11-s − 1.51i·12-s − 1.30i·13-s + (−0.664 − 1.78i)14-s − 0.258·15-s + 3.23·16-s − 0.765i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.349 + 0.936i$ motivic weight = $$2$$ character : $\chi_{105} (76, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1),\ 0.349 + 0.936i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.497459 - 0.345394i$$ $$L(\frac12)$$ $$\approx$$ $$0.497459 - 0.345394i$$ $$L(2)$$ 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 + 1.73iT$$
5 $$1 + 2.23iT$$
7 $$1 + (-2.44 - 6.55i)T$$
good2 $$1 + 3.80T + 4T^{2}$$
11 $$1 - 14.4T + 121T^{2}$$
13 $$1 + 16.9iT - 169T^{2}$$
17 $$1 + 13.0iT - 289T^{2}$$
19 $$1 + 18.6iT - 361T^{2}$$
23 $$1 - 10.3T + 529T^{2}$$
29 $$1 + 13.7T + 841T^{2}$$
31 $$1 + 42.4iT - 961T^{2}$$
37 $$1 - 28.7T + 1.36e3T^{2}$$
41 $$1 - 28.8iT - 1.68e3T^{2}$$
43 $$1 + 5.84T + 1.84e3T^{2}$$
47 $$1 + 10.5iT - 2.20e3T^{2}$$
53 $$1 - 81.9T + 2.80e3T^{2}$$
59 $$1 + 35.1iT - 3.48e3T^{2}$$
61 $$1 - 68.4iT - 3.72e3T^{2}$$
67 $$1 - 47.4T + 4.48e3T^{2}$$
71 $$1 - 47.6T + 5.04e3T^{2}$$
73 $$1 - 125. iT - 5.32e3T^{2}$$
79 $$1 + 129.T + 6.24e3T^{2}$$
83 $$1 - 42.6iT - 6.88e3T^{2}$$
89 $$1 + 25.5iT - 7.92e3T^{2}$$
97 $$1 + 28.7iT - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}