# 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 − 3.82·2-s − 3·3-s + 6.65·4-s + 5·5-s + 11.4·6-s − 7·7-s + 5.14·8-s + 9·9-s − 19.1·10-s + 48.5·11-s − 19.9·12-s − 43.6·13-s + 26.7·14-s − 15·15-s − 72.9·16-s − 67.6·17-s − 34.4·18-s − 93.2·19-s + 33.2·20-s + 21·21-s − 185.·22-s − 104.·23-s − 15.4·24-s + 25·25-s + 167.·26-s − 27·27-s − 46.5·28-s + ⋯
 L(s)  = 1 − 1.35·2-s − 0.577·3-s + 0.832·4-s + 0.447·5-s + 0.781·6-s − 0.377·7-s + 0.227·8-s + 0.333·9-s − 0.605·10-s + 1.33·11-s − 0.480·12-s − 0.931·13-s + 0.511·14-s − 0.258·15-s − 1.13·16-s − 0.965·17-s − 0.451·18-s − 1.12·19-s + 0.372·20-s + 0.218·21-s − 1.80·22-s − 0.944·23-s − 0.131·24-s + 0.200·25-s + 1.26·26-s − 0.192·27-s − 0.314·28-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 + 3.82T + 8T^{2}$$
11 $$1 - 48.5T + 1.33e3T^{2}$$
13 $$1 + 43.6T + 2.19e3T^{2}$$
17 $$1 + 67.6T + 4.91e3T^{2}$$
19 $$1 + 93.2T + 6.85e3T^{2}$$
23 $$1 + 104.T + 1.21e4T^{2}$$
29 $$1 + 58.7T + 2.43e4T^{2}$$
31 $$1 + 9.08T + 2.97e4T^{2}$$
37 $$1 + 252.T + 5.06e4T^{2}$$
41 $$1 - 276.T + 6.89e4T^{2}$$
43 $$1 + 92.6T + 7.95e4T^{2}$$
47 $$1 + 582.T + 1.03e5T^{2}$$
53 $$1 - 623.T + 1.48e5T^{2}$$
59 $$1 + 524.T + 2.05e5T^{2}$$
61 $$1 + 352.T + 2.26e5T^{2}$$
67 $$1 + 736.T + 3.00e5T^{2}$$
71 $$1 + 492.T + 3.57e5T^{2}$$
73 $$1 - 1.16e3T + 3.89e5T^{2}$$
79 $$1 + 872.T + 4.93e5T^{2}$$
83 $$1 + 529.T + 5.71e5T^{2}$$
89 $$1 + 385.T + 7.04e5T^{2}$$
97 $$1 + 463.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}