# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.70·2-s + 3·3-s − 5.10·4-s − 5·5-s − 5.10·6-s + 7·7-s + 22.2·8-s + 9·9-s + 8.50·10-s + 37.4·11-s − 15.3·12-s + 29.0·13-s − 11.9·14-s − 15·15-s + 2.89·16-s + 58.4·17-s − 15.3·18-s − 54.5·19-s + 25.5·20-s + 21·21-s − 63.6·22-s + 161.·23-s + 66.8·24-s + 25·25-s − 49.3·26-s + 27·27-s − 35.7·28-s + ⋯
 L(s)  = 1 − 0.601·2-s + 0.577·3-s − 0.638·4-s − 0.447·5-s − 0.347·6-s + 0.377·7-s + 0.985·8-s + 0.333·9-s + 0.269·10-s + 1.02·11-s − 0.368·12-s + 0.619·13-s − 0.227·14-s − 0.258·15-s + 0.0452·16-s + 0.833·17-s − 0.200·18-s − 0.659·19-s + 0.285·20-s + 0.218·21-s − 0.616·22-s + 1.46·23-s + 0.568·24-s + 0.200·25-s − 0.372·26-s + 0.192·27-s − 0.241·28-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 = $$0$$ Selberg data = $$(2,\ 105,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$1.25861$$ $$L(\frac12)$$ $$\approx$$ $$1.25861$$ $$L(\frac{5}{2})$$ 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 + 1.70T + 8T^{2}$$
11 $$1 - 37.4T + 1.33e3T^{2}$$
13 $$1 - 29.0T + 2.19e3T^{2}$$
17 $$1 - 58.4T + 4.91e3T^{2}$$
19 $$1 + 54.5T + 6.85e3T^{2}$$
23 $$1 - 161.T + 1.21e4T^{2}$$
29 $$1 - 137.T + 2.43e4T^{2}$$
31 $$1 - 154.T + 2.97e4T^{2}$$
37 $$1 + 350.T + 5.06e4T^{2}$$
41 $$1 - 353.T + 6.89e4T^{2}$$
43 $$1 + 518.T + 7.95e4T^{2}$$
47 $$1 + 542.T + 1.03e5T^{2}$$
53 $$1 - 305.T + 1.48e5T^{2}$$
59 $$1 - 14.6T + 2.05e5T^{2}$$
61 $$1 + 171.T + 2.26e5T^{2}$$
67 $$1 - 551.T + 3.00e5T^{2}$$
71 $$1 + 120.T + 3.57e5T^{2}$$
73 $$1 - 284.T + 3.89e5T^{2}$$
79 $$1 - 941.T + 4.93e5T^{2}$$
83 $$1 - 377.T + 5.71e5T^{2}$$
89 $$1 + 677.T + 7.04e5T^{2}$$
97 $$1 + 1.22e3T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}