# Properties

 Label 2-12e3-1.1-c3-0-30 Degree $2$ Conductor $1728$ Sign $1$ Analytic cond. $101.955$ Root an. cond. $10.0972$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 15.4·5-s + 23.8·7-s + 14.2·11-s + 13.8·13-s + 80.5·17-s + 144.·19-s + 141.·23-s + 112.·25-s − 251.·29-s − 16.6·31-s − 367.·35-s − 305.·37-s − 429.·41-s + 181.·43-s + 79.4·47-s + 225.·49-s − 663.·53-s − 219.·55-s + 220.·59-s + 473.·61-s − 213.·65-s + 647.·67-s − 14.4·71-s + 776.·73-s + 339.·77-s − 257.·79-s + 1.28e3·83-s + ⋯
 L(s)  = 1 − 1.37·5-s + 1.28·7-s + 0.390·11-s + 0.295·13-s + 1.14·17-s + 1.74·19-s + 1.27·23-s + 0.901·25-s − 1.60·29-s − 0.0965·31-s − 1.77·35-s − 1.35·37-s − 1.63·41-s + 0.644·43-s + 0.246·47-s + 0.655·49-s − 1.72·53-s − 0.538·55-s + 0.486·59-s + 0.993·61-s − 0.406·65-s + 1.18·67-s − 0.0242·71-s + 1.24·73-s + 0.502·77-s − 0.367·79-s + 1.69·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $1$ Analytic conductor: $$101.955$$ Root analytic conductor: $$10.0972$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.208113255$$ $$L(\frac12)$$ $$\approx$$ $$2.208113255$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 15.4T + 125T^{2}$$
7 $$1 - 23.8T + 343T^{2}$$
11 $$1 - 14.2T + 1.33e3T^{2}$$
13 $$1 - 13.8T + 2.19e3T^{2}$$
17 $$1 - 80.5T + 4.91e3T^{2}$$
19 $$1 - 144.T + 6.85e3T^{2}$$
23 $$1 - 141.T + 1.21e4T^{2}$$
29 $$1 + 251.T + 2.43e4T^{2}$$
31 $$1 + 16.6T + 2.97e4T^{2}$$
37 $$1 + 305.T + 5.06e4T^{2}$$
41 $$1 + 429.T + 6.89e4T^{2}$$
43 $$1 - 181.T + 7.95e4T^{2}$$
47 $$1 - 79.4T + 1.03e5T^{2}$$
53 $$1 + 663.T + 1.48e5T^{2}$$
59 $$1 - 220.T + 2.05e5T^{2}$$
61 $$1 - 473.T + 2.26e5T^{2}$$
67 $$1 - 647.T + 3.00e5T^{2}$$
71 $$1 + 14.4T + 3.57e5T^{2}$$
73 $$1 - 776.T + 3.89e5T^{2}$$
79 $$1 + 257.T + 4.93e5T^{2}$$
83 $$1 - 1.28e3T + 5.71e5T^{2}$$
89 $$1 - 156.T + 7.04e5T^{2}$$
97 $$1 - 1.16e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$