# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $-0.146 - 0.989i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.31 + 1.91i)2-s + (−2.97 + 0.410i)3-s + (5.34 + 9.24i)4-s + (1.93 + 1.11i)5-s + (−10.6 − 4.32i)6-s + (−5.86 − 3.82i)7-s + 25.5i·8-s + (8.66 − 2.44i)9-s + (4.28 + 7.41i)10-s + (9.48 − 5.47i)11-s + (−19.6 − 25.2i)12-s − 3.75·13-s + (−12.1 − 23.9i)14-s + (−6.21 − 2.52i)15-s + (−27.6 + 47.9i)16-s + (12.6 − 7.30i)17-s + ⋯
 L(s)  = 1 + (1.65 + 0.957i)2-s + (−0.990 + 0.136i)3-s + (1.33 + 2.31i)4-s + (0.387 + 0.223i)5-s + (−1.77 − 0.721i)6-s + (−0.837 − 0.546i)7-s + 3.19i·8-s + (0.962 − 0.271i)9-s + (0.428 + 0.741i)10-s + (0.862 − 0.497i)11-s + (−1.63 − 2.10i)12-s − 0.288·13-s + (−0.866 − 1.70i)14-s + (−0.414 − 0.168i)15-s + (−1.72 + 2.99i)16-s + (0.744 − 0.429i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.146 - 0.989i$ motivic weight = $$2$$ character : $\chi_{105} (86, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1),\ -0.146 - 0.989i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.60892 + 1.86406i$$ $$L(\frac12)$$ $$\approx$$ $$1.60892 + 1.86406i$$ $$L(2)$$ 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 + (2.97 - 0.410i)T$$
5 $$1 + (-1.93 - 1.11i)T$$
7 $$1 + (5.86 + 3.82i)T$$
good2 $$1 + (-3.31 - 1.91i)T + (2 + 3.46i)T^{2}$$
11 $$1 + (-9.48 + 5.47i)T + (60.5 - 104. i)T^{2}$$
13 $$1 + 3.75T + 169T^{2}$$
17 $$1 + (-12.6 + 7.30i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (-5.54 + 9.60i)T + (-180.5 - 312. i)T^{2}$$
23 $$1 + (8.53 + 4.92i)T + (264.5 + 458. i)T^{2}$$
29 $$1 + 10.0iT - 841T^{2}$$
31 $$1 + (12.0 + 20.9i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 + (-19.1 + 33.2i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 - 67.5iT - 1.68e3T^{2}$$
43 $$1 + 77.4T + 1.84e3T^{2}$$
47 $$1 + (-4.91 - 2.83i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (59.9 - 34.6i)T + (1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (45.4 - 26.2i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-6.77 + 11.7i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (10.6 + 18.4i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + 25.6iT - 5.04e3T^{2}$$
73 $$1 + (-12.4 - 21.4i)T + (-2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-55.3 + 95.8i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 - 102. iT - 6.88e3T^{2}$$
89 $$1 + (62.5 + 36.1i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + 6.23T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}