# Properties

 Label 2-675-1.1-c3-0-47 Degree $2$ Conductor $675$ Sign $-1$ Analytic cond. $39.8262$ Root an. cond. $6.31080$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.258·2-s − 7.93·4-s − 14.5·7-s + 4.12·8-s + 49.2·11-s − 72.1·13-s + 3.75·14-s + 62.3·16-s + 118.·17-s + 123.·19-s − 12.7·22-s − 91.4·23-s + 18.6·26-s + 115.·28-s − 174.·29-s − 46.2·31-s − 49.1·32-s − 30.5·34-s − 154.·37-s − 31.9·38-s + 364.·41-s − 125.·43-s − 390.·44-s + 23.6·46-s − 221.·47-s − 132.·49-s + 572.·52-s + ⋯
 L(s)  = 1 − 0.0914·2-s − 0.991·4-s − 0.783·7-s + 0.182·8-s + 1.35·11-s − 1.53·13-s + 0.0716·14-s + 0.974·16-s + 1.68·17-s + 1.48·19-s − 0.123·22-s − 0.829·23-s + 0.140·26-s + 0.777·28-s − 1.11·29-s − 0.268·31-s − 0.271·32-s − 0.154·34-s − 0.688·37-s − 0.136·38-s + 1.38·41-s − 0.445·43-s − 1.33·44-s + 0.0758·46-s − 0.687·47-s − 0.385·49-s + 1.52·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$675$$    =    $$3^{3} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$39.8262$$ Root analytic conductor: $$6.31080$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{675} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 675,\ (\ :3/2),\ -1)$$

## 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$$
good2 $$1 + 0.258T + 8T^{2}$$
7 $$1 + 14.5T + 343T^{2}$$
11 $$1 - 49.2T + 1.33e3T^{2}$$
13 $$1 + 72.1T + 2.19e3T^{2}$$
17 $$1 - 118.T + 4.91e3T^{2}$$
19 $$1 - 123.T + 6.85e3T^{2}$$
23 $$1 + 91.4T + 1.21e4T^{2}$$
29 $$1 + 174.T + 2.43e4T^{2}$$
31 $$1 + 46.2T + 2.97e4T^{2}$$
37 $$1 + 154.T + 5.06e4T^{2}$$
41 $$1 - 364.T + 6.89e4T^{2}$$
43 $$1 + 125.T + 7.95e4T^{2}$$
47 $$1 + 221.T + 1.03e5T^{2}$$
53 $$1 + 13.6T + 1.48e5T^{2}$$
59 $$1 + 239.T + 2.05e5T^{2}$$
61 $$1 + 54.5T + 2.26e5T^{2}$$
67 $$1 - 76.0T + 3.00e5T^{2}$$
71 $$1 + 728.T + 3.57e5T^{2}$$
73 $$1 - 501.T + 3.89e5T^{2}$$
79 $$1 - 397.T + 4.93e5T^{2}$$
83 $$1 + 1.36e3T + 5.71e5T^{2}$$
89 $$1 + 1.46e3T + 7.04e5T^{2}$$
97 $$1 + 335.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$