# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.23·2-s + 3·3-s + 9.94·4-s − 5·5-s − 12.7·6-s − 7·7-s − 8.23·8-s + 9·9-s + 21.1·10-s − 41.5·11-s + 29.8·12-s + 88.9·13-s + 29.6·14-s − 15·15-s − 44.6·16-s − 120.·17-s − 38.1·18-s − 112.·19-s − 49.7·20-s − 21·21-s + 175.·22-s − 115.·23-s − 24.7·24-s + 25·25-s − 376.·26-s + 27·27-s − 69.6·28-s + ⋯
 L(s)  = 1 − 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.447·5-s − 0.864·6-s − 0.377·7-s − 0.363·8-s + 0.333·9-s + 0.669·10-s − 1.13·11-s + 0.717·12-s + 1.89·13-s + 0.566·14-s − 0.258·15-s − 0.697·16-s − 1.71·17-s − 0.499·18-s − 1.35·19-s − 0.555·20-s − 0.218·21-s + 1.70·22-s − 1.04·23-s − 0.210·24-s + 0.200·25-s − 2.84·26-s + 0.192·27-s − 0.469·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$3$$ character : $\chi_{105} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 105,\ (\ :3/2),\ -1)$$ $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 3T$$
5 $$1 + 5T$$
7 $$1 + 7T$$
good2 $$1 + 4.23T + 8T^{2}$$
11 $$1 + 41.5T + 1.33e3T^{2}$$
13 $$1 - 88.9T + 2.19e3T^{2}$$
17 $$1 + 120.T + 4.91e3T^{2}$$
19 $$1 + 112.T + 6.85e3T^{2}$$
23 $$1 + 115.T + 1.21e4T^{2}$$
29 $$1 + 144.T + 2.43e4T^{2}$$
31 $$1 + 258.T + 2.97e4T^{2}$$
37 $$1 - 48.3T + 5.06e4T^{2}$$
41 $$1 - 200.T + 6.89e4T^{2}$$
43 $$1 + 218.T + 7.95e4T^{2}$$
47 $$1 - 575.T + 1.03e5T^{2}$$
53 $$1 + 184.T + 1.48e5T^{2}$$
59 $$1 + 151.T + 2.05e5T^{2}$$
61 $$1 + 529.T + 2.26e5T^{2}$$
67 $$1 - 1.28T + 3.00e5T^{2}$$
71 $$1 + 61.4T + 3.57e5T^{2}$$
73 $$1 - 484.T + 3.89e5T^{2}$$
79 $$1 - 878.T + 4.93e5T^{2}$$
83 $$1 - 491.T + 5.71e5T^{2}$$
89 $$1 + 415.T + 7.04e5T^{2}$$
97 $$1 + 1.03e3T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}