# Properties

 Label 2-1216-19.18-c2-0-35 Degree $2$ Conductor $1216$ Sign $0.315 + 0.948i$ Analytic cond. $33.1336$ Root an. cond. $5.75617$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.60i·3-s − 4·5-s − 5·7-s − 3.99·9-s + 10·11-s + 3.60i·13-s + 14.4i·15-s + 15·17-s + (6 + 18.0i)19-s + 18.0i·21-s + 35·23-s − 9·25-s − 18.0i·27-s + 18.0i·29-s + 36.0i·31-s + ⋯
 L(s)  = 1 − 1.20i·3-s − 0.800·5-s − 0.714·7-s − 0.444·9-s + 0.909·11-s + 0.277i·13-s + 0.961i·15-s + 0.882·17-s + (0.315 + 0.948i)19-s + 0.858i·21-s + 1.52·23-s − 0.359·25-s − 0.667i·27-s + 0.621i·29-s + 1.16i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $0.315 + 0.948i$ Analytic conductor: $$33.1336$$ Root analytic conductor: $$5.75617$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (1025, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :1),\ 0.315 + 0.948i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.616439011$$ $$L(\frac12)$$ $$\approx$$ $$1.616439011$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (-6 - 18.0i)T$$
good3 $$1 + 3.60iT - 9T^{2}$$
5 $$1 + 4T + 25T^{2}$$
7 $$1 + 5T + 49T^{2}$$
11 $$1 - 10T + 121T^{2}$$
13 $$1 - 3.60iT - 169T^{2}$$
17 $$1 - 15T + 289T^{2}$$
23 $$1 - 35T + 529T^{2}$$
29 $$1 - 18.0iT - 841T^{2}$$
31 $$1 - 36.0iT - 961T^{2}$$
37 $$1 + 21.6iT - 1.36e3T^{2}$$
41 $$1 + 36.0iT - 1.68e3T^{2}$$
43 $$1 - 20T + 1.84e3T^{2}$$
47 $$1 - 10T + 2.20e3T^{2}$$
53 $$1 + 75.7iT - 2.80e3T^{2}$$
59 $$1 - 18.0iT - 3.48e3T^{2}$$
61 $$1 - 40T + 3.72e3T^{2}$$
67 $$1 - 39.6iT - 4.48e3T^{2}$$
71 $$1 + 108. iT - 5.04e3T^{2}$$
73 $$1 - 105T + 5.32e3T^{2}$$
79 $$1 - 36.0iT - 6.24e3T^{2}$$
83 $$1 - 40T + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 + 122. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$