# Properties

 Label 2-1575-1.1-c3-0-50 Degree $2$ Conductor $1575$ Sign $1$ Analytic cond. $92.9280$ Root an. cond. $9.63991$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.82·2-s − 4.65·4-s + 7·7-s − 23.1·8-s + 64.5·11-s + 32.3·13-s + 12.7·14-s − 5.05·16-s − 56.3·17-s − 2.74·19-s + 118.·22-s + 88.1·23-s + 59.1·26-s − 32.5·28-s − 246.·29-s − 110.·31-s + 175.·32-s − 103.·34-s − 120.·37-s − 5.01·38-s + 176.·41-s + 443.·43-s − 300.·44-s + 161.·46-s − 345.·47-s + 49·49-s − 150.·52-s + ⋯
 L(s)  = 1 + 0.646·2-s − 0.582·4-s + 0.377·7-s − 1.02·8-s + 1.76·11-s + 0.690·13-s + 0.244·14-s − 0.0790·16-s − 0.803·17-s − 0.0331·19-s + 1.14·22-s + 0.799·23-s + 0.446·26-s − 0.220·28-s − 1.57·29-s − 0.642·31-s + 0.971·32-s − 0.519·34-s − 0.536·37-s − 0.0214·38-s + 0.671·41-s + 1.57·43-s − 1.03·44-s + 0.516·46-s − 1.07·47-s + 0.142·49-s − 0.401·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1575$$    =    $$3^{2} \cdot 5^{2} \cdot 7$$ Sign: $1$ Analytic conductor: $$92.9280$$ Root analytic conductor: $$9.63991$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1575} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1575,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.764164022$$ $$L(\frac12)$$ $$\approx$$ $$2.764164022$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1 - 7T$$
good2 $$1 - 1.82T + 8T^{2}$$
11 $$1 - 64.5T + 1.33e3T^{2}$$
13 $$1 - 32.3T + 2.19e3T^{2}$$
17 $$1 + 56.3T + 4.91e3T^{2}$$
19 $$1 + 2.74T + 6.85e3T^{2}$$
23 $$1 - 88.1T + 1.21e4T^{2}$$
29 $$1 + 246.T + 2.43e4T^{2}$$
31 $$1 + 110.T + 2.97e4T^{2}$$
37 $$1 + 120.T + 5.06e4T^{2}$$
41 $$1 - 176.T + 6.89e4T^{2}$$
43 $$1 - 443.T + 7.95e4T^{2}$$
47 $$1 + 345.T + 1.03e5T^{2}$$
53 $$1 - 260.T + 1.48e5T^{2}$$
59 $$1 + 628.T + 2.05e5T^{2}$$
61 $$1 + 115.T + 2.26e5T^{2}$$
67 $$1 - 951.T + 3.00e5T^{2}$$
71 $$1 + 356.T + 3.57e5T^{2}$$
73 $$1 - 656.T + 3.89e5T^{2}$$
79 $$1 - 440.T + 4.93e5T^{2}$$
83 $$1 + 54.4T + 5.71e5T^{2}$$
89 $$1 - 1.01e3T + 7.04e5T^{2}$$
97 $$1 - 724.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$