# Properties

 Label 2-1575-1.1-c3-0-105 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.561·2-s − 7.68·4-s + 7·7-s + 8.80·8-s + 25.8·11-s + 15.5·13-s − 3.93·14-s + 56.5·16-s − 95.1·17-s − 143.·19-s − 14.4·22-s + 77.5·23-s − 8.73·26-s − 53.7·28-s + 204.·29-s − 40.6·31-s − 102.·32-s + 53.4·34-s − 95.6·37-s + 80.5·38-s − 24.7·41-s − 282.·43-s − 198.·44-s − 43.5·46-s + 257.·47-s + 49·49-s − 119.·52-s + ⋯
 L(s)  = 1 − 0.198·2-s − 0.960·4-s + 0.377·7-s + 0.389·8-s + 0.707·11-s + 0.331·13-s − 0.0750·14-s + 0.883·16-s − 1.35·17-s − 1.73·19-s − 0.140·22-s + 0.702·23-s − 0.0658·26-s − 0.363·28-s + 1.30·29-s − 0.235·31-s − 0.564·32-s + 0.269·34-s − 0.425·37-s + 0.343·38-s − 0.0941·41-s − 1.00·43-s − 0.679·44-s − 0.139·46-s + 0.798·47-s + 0.142·49-s − 0.318·52-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: $$1$$ Selberg data: $$(2,\ 1575,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 0.561T + 8T^{2}$$
11 $$1 - 25.8T + 1.33e3T^{2}$$
13 $$1 - 15.5T + 2.19e3T^{2}$$
17 $$1 + 95.1T + 4.91e3T^{2}$$
19 $$1 + 143.T + 6.85e3T^{2}$$
23 $$1 - 77.5T + 1.21e4T^{2}$$
29 $$1 - 204.T + 2.43e4T^{2}$$
31 $$1 + 40.6T + 2.97e4T^{2}$$
37 $$1 + 95.6T + 5.06e4T^{2}$$
41 $$1 + 24.7T + 6.89e4T^{2}$$
43 $$1 + 282.T + 7.95e4T^{2}$$
47 $$1 - 257.T + 1.03e5T^{2}$$
53 $$1 - 257.T + 1.48e5T^{2}$$
59 $$1 - 651.T + 2.05e5T^{2}$$
61 $$1 - 451.T + 2.26e5T^{2}$$
67 $$1 + 832.T + 3.00e5T^{2}$$
71 $$1 - 174.T + 3.57e5T^{2}$$
73 $$1 - 47.4T + 3.89e5T^{2}$$
79 $$1 + 1.16T + 4.93e5T^{2}$$
83 $$1 - 1.49e3T + 5.71e5T^{2}$$
89 $$1 + 1.30e3T + 7.04e5T^{2}$$
97 $$1 - 1.36e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$