# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $0.935 - 0.353i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.67 − 1.67i)2-s + (−1.56 + 2.55i)3-s − 1.58i·4-s + (0.529 + 4.97i)5-s + (1.66 + 6.89i)6-s + (6.78 − 1.73i)7-s + (4.03 + 4.03i)8-s + (−4.10 − 8.01i)9-s + (9.19 + 7.42i)10-s + 4.41i·11-s + (4.06 + 2.48i)12-s + (1.62 + 1.62i)13-s + (8.43 − 14.2i)14-s + (−13.5 − 6.42i)15-s + 19.8·16-s + (−13.9 − 13.9i)17-s + ⋯
 L(s)  = 1 + (0.835 − 0.835i)2-s + (−0.521 + 0.853i)3-s − 0.397i·4-s + (0.105 + 0.994i)5-s + (0.277 + 1.14i)6-s + (0.968 − 0.247i)7-s + (0.503 + 0.503i)8-s + (−0.455 − 0.890i)9-s + (0.919 + 0.742i)10-s + 0.401i·11-s + (0.338 + 0.207i)12-s + (0.124 + 0.124i)13-s + (0.602 − 1.01i)14-s + (−0.903 − 0.428i)15-s + 1.23·16-s + (−0.819 − 0.819i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.935 - 0.353i$ motivic weight = $$2$$ character : $\chi_{105} (62, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1),\ 0.935 - 0.353i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.84532 + 0.337128i$$ $$L(\frac12)$$ $$\approx$$ $$1.84532 + 0.337128i$$ $$L(2)$$ 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 + (1.56 - 2.55i)T$$
5 $$1 + (-0.529 - 4.97i)T$$
7 $$1 + (-6.78 + 1.73i)T$$
good2 $$1 + (-1.67 + 1.67i)T - 4iT^{2}$$
11 $$1 - 4.41iT - 121T^{2}$$
13 $$1 + (-1.62 - 1.62i)T + 169iT^{2}$$
17 $$1 + (13.9 + 13.9i)T + 289iT^{2}$$
19 $$1 + 0.694T + 361T^{2}$$
23 $$1 + (23.1 + 23.1i)T + 529iT^{2}$$
29 $$1 - 49.1T + 841T^{2}$$
31 $$1 + 33.8iT - 961T^{2}$$
37 $$1 + (-2.02 - 2.02i)T + 1.36e3iT^{2}$$
41 $$1 - 32.5T + 1.68e3T^{2}$$
43 $$1 + (30.4 - 30.4i)T - 1.84e3iT^{2}$$
47 $$1 + (18.7 + 18.7i)T + 2.20e3iT^{2}$$
53 $$1 + (33.8 + 33.8i)T + 2.80e3iT^{2}$$
59 $$1 + 23.1iT - 3.48e3T^{2}$$
61 $$1 - 12.9iT - 3.72e3T^{2}$$
67 $$1 + (56.3 + 56.3i)T + 4.48e3iT^{2}$$
71 $$1 + 92.7iT - 5.04e3T^{2}$$
73 $$1 + (-95.4 - 95.4i)T + 5.32e3iT^{2}$$
79 $$1 - 100. iT - 6.24e3T^{2}$$
83 $$1 + (-5.62 + 5.62i)T - 6.88e3iT^{2}$$
89 $$1 - 158. iT - 7.92e3T^{2}$$
97 $$1 + (37.1 - 37.1i)T - 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}