# Properties

 Label 2-546-1.1-c7-0-68 Degree $2$ Conductor $546$ Sign $-1$ Analytic cond. $170.562$ Root an. cond. $13.0599$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8·2-s + 27·3-s + 64·4-s − 12.1·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 97.2·10-s + 2.35e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 328.·15-s + 4.09e3·16-s − 2.37e4·17-s − 5.83e3·18-s + 4.09e4·19-s − 777.·20-s + 9.26e3·21-s − 1.88e4·22-s − 3.67e4·23-s − 1.38e4·24-s − 7.79e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0434·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.0307·10-s + 0.532·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0250·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 1.36·19-s − 0.0217·20-s + 0.218·21-s − 0.376·22-s − 0.629·23-s − 0.204·24-s − 0.998·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-1$ Analytic conductor: $$170.562$$ Root analytic conductor: $$13.0599$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{546} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 546,\ (\ :7/2),\ -1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 8T$$
3 $$1 - 27T$$
7 $$1 - 343T$$
13 $$1 - 2.19e3T$$
good5 $$1 + 12.1T + 7.81e4T^{2}$$
11 $$1 - 2.35e3T + 1.94e7T^{2}$$
17 $$1 + 2.37e4T + 4.10e8T^{2}$$
19 $$1 - 4.09e4T + 8.93e8T^{2}$$
23 $$1 + 3.67e4T + 3.40e9T^{2}$$
29 $$1 + 1.45e5T + 1.72e10T^{2}$$
31 $$1 + 9.10e4T + 2.75e10T^{2}$$
37 $$1 - 3.89e5T + 9.49e10T^{2}$$
41 $$1 - 5.92e5T + 1.94e11T^{2}$$
43 $$1 + 4.08e5T + 2.71e11T^{2}$$
47 $$1 + 1.19e6T + 5.06e11T^{2}$$
53 $$1 + 1.59e6T + 1.17e12T^{2}$$
59 $$1 - 6.97e5T + 2.48e12T^{2}$$
61 $$1 + 2.63e6T + 3.14e12T^{2}$$
67 $$1 - 2.94e6T + 6.06e12T^{2}$$
71 $$1 - 2.79e6T + 9.09e12T^{2}$$
73 $$1 - 3.60e6T + 1.10e13T^{2}$$
79 $$1 + 5.23e6T + 1.92e13T^{2}$$
83 $$1 + 3.05e6T + 2.71e13T^{2}$$
89 $$1 + 2.20e6T + 4.42e13T^{2}$$
97 $$1 + 1.71e7T + 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$