# Properties

 Label 2-1216-1.1-c1-0-15 Degree $2$ Conductor $1216$ Sign $-1$ Analytic cond. $9.70980$ Root an. cond. $3.11605$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s − 7-s + 6·9-s − 2·11-s + 13-s + 3·17-s + 19-s + 3·21-s + 3·23-s − 5·25-s − 9·27-s − 3·29-s + 8·31-s + 6·33-s + 10·37-s − 3·39-s − 12·41-s − 8·43-s − 8·47-s − 6·49-s − 9·51-s + 9·53-s − 3·57-s + 5·59-s − 10·61-s − 6·63-s − 7·67-s + ⋯
 L(s)  = 1 − 1.73·3-s − 0.377·7-s + 2·9-s − 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.229·19-s + 0.654·21-s + 0.625·23-s − 25-s − 1.73·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s + 1.64·37-s − 0.480·39-s − 1.87·41-s − 1.21·43-s − 1.16·47-s − 6/7·49-s − 1.26·51-s + 1.23·53-s − 0.397·57-s + 0.650·59-s − 1.28·61-s − 0.755·63-s − 0.855·67-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $-1$ Analytic conductor: $$9.70980$$ Root analytic conductor: $$3.11605$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1216,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 - T$$
good3 $$1 + p T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
23 $$1 - 3 T + p T^{2}$$
29 $$1 + 3 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 + 12 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 9 T + p T^{2}$$
59 $$1 - 5 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + 7 T + p T^{2}$$
71 $$1 + 10 T + p T^{2}$$
73 $$1 - T + p T^{2}$$
79 $$1 + 14 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + 4 T + p T^{2}$$
97 $$1 + 6 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$