# Properties

 Label 2-1368-1.1-c3-0-19 Degree $2$ Conductor $1368$ Sign $1$ Analytic cond. $80.7146$ Root an. cond. $8.98413$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 13.8·5-s − 9.22·7-s − 59.8·11-s + 38.2·13-s + 47.0·17-s − 19·19-s − 179.·23-s + 66.1·25-s + 96.6·29-s + 125.·31-s − 127.·35-s + 407.·37-s + 220.·41-s + 100.·43-s + 213.·47-s − 257.·49-s − 19.8·53-s − 827.·55-s + 97.7·59-s + 266.·61-s + 528.·65-s + 627.·67-s + 644.·71-s + 807.·73-s + 552.·77-s + 785.·79-s − 1.17e3·83-s + ⋯
 L(s)  = 1 + 1.23·5-s − 0.497·7-s − 1.64·11-s + 0.815·13-s + 0.671·17-s − 0.229·19-s − 1.62·23-s + 0.529·25-s + 0.618·29-s + 0.729·31-s − 0.615·35-s + 1.81·37-s + 0.840·41-s + 0.357·43-s + 0.663·47-s − 0.752·49-s − 0.0513·53-s − 2.02·55-s + 0.215·59-s + 0.559·61-s + 1.00·65-s + 1.14·67-s + 1.07·71-s + 1.29·73-s + 0.817·77-s + 1.11·79-s − 1.55·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1368$$    =    $$2^{3} \cdot 3^{2} \cdot 19$$ Sign: $1$ Analytic conductor: $$80.7146$$ Root analytic conductor: $$8.98413$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1368,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.346241156$$ $$L(\frac12)$$ $$\approx$$ $$2.346241156$$ $$L(\frac{5}{2})$$ 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$$
19 $$1 + 19T$$
good5 $$1 - 13.8T + 125T^{2}$$
7 $$1 + 9.22T + 343T^{2}$$
11 $$1 + 59.8T + 1.33e3T^{2}$$
13 $$1 - 38.2T + 2.19e3T^{2}$$
17 $$1 - 47.0T + 4.91e3T^{2}$$
23 $$1 + 179.T + 1.21e4T^{2}$$
29 $$1 - 96.6T + 2.43e4T^{2}$$
31 $$1 - 125.T + 2.97e4T^{2}$$
37 $$1 - 407.T + 5.06e4T^{2}$$
41 $$1 - 220.T + 6.89e4T^{2}$$
43 $$1 - 100.T + 7.95e4T^{2}$$
47 $$1 - 213.T + 1.03e5T^{2}$$
53 $$1 + 19.8T + 1.48e5T^{2}$$
59 $$1 - 97.7T + 2.05e5T^{2}$$
61 $$1 - 266.T + 2.26e5T^{2}$$
67 $$1 - 627.T + 3.00e5T^{2}$$
71 $$1 - 644.T + 3.57e5T^{2}$$
73 $$1 - 807.T + 3.89e5T^{2}$$
79 $$1 - 785.T + 4.93e5T^{2}$$
83 $$1 + 1.17e3T + 5.71e5T^{2}$$
89 $$1 - 234.T + 7.04e5T^{2}$$
97 $$1 + 1.34e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$