# Properties

 Label 2-1216-8.5-c1-0-14 Degree $2$ Conductor $1216$ Sign $0.965 + 0.258i$ Analytic cond. $9.70980$ Root an. cond. $3.11605$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.04i·5-s + 0.418·7-s + 3·9-s + 5.27i·11-s + 2.62i·13-s + 3.27·17-s + i·19-s + 3.46·23-s − 4.27·25-s + 2.62i·29-s + 6.09·31-s − 1.27i·35-s − 9.55i·37-s + 4.54·41-s + 2.72i·43-s + ⋯
 L(s)  = 1 − 1.36i·5-s + 0.158·7-s + 9-s + 1.59i·11-s + 0.728i·13-s + 0.794·17-s + 0.229i·19-s + 0.722·23-s − 0.854·25-s + 0.487i·29-s + 1.09·31-s − 0.215i·35-s − 1.57i·37-s + 0.710·41-s + 0.415i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $0.965 + 0.258i$ Analytic conductor: $$9.70980$$ Root analytic conductor: $$3.11605$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (609, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :1/2),\ 0.965 + 0.258i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.882364678$$ $$L(\frac12)$$ $$\approx$$ $$1.882364678$$ $$L(\frac{3}{2})$$ 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 - iT$$
good3 $$1 - 3T^{2}$$
5 $$1 + 3.04iT - 5T^{2}$$
7 $$1 - 0.418T + 7T^{2}$$
11 $$1 - 5.27iT - 11T^{2}$$
13 $$1 - 2.62iT - 13T^{2}$$
17 $$1 - 3.27T + 17T^{2}$$
23 $$1 - 3.46T + 23T^{2}$$
29 $$1 - 2.62iT - 29T^{2}$$
31 $$1 - 6.09T + 31T^{2}$$
37 $$1 + 9.55iT - 37T^{2}$$
41 $$1 - 4.54T + 41T^{2}$$
43 $$1 - 2.72iT - 43T^{2}$$
47 $$1 + 0.418T + 47T^{2}$$
53 $$1 + 10.3iT - 53T^{2}$$
59 $$1 + 6.54iT - 59T^{2}$$
61 $$1 + 3.04iT - 61T^{2}$$
67 $$1 + 6.54iT - 67T^{2}$$
71 $$1 + 11.3T + 71T^{2}$$
73 $$1 + 3.27T + 73T^{2}$$
79 $$1 - 13.0T + 79T^{2}$$
83 $$1 - 17.0iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 - 16.5T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$