# Properties

 Label 2-1216-1.1-c3-0-106 Degree $2$ Conductor $1216$ Sign $-1$ Analytic cond. $71.7463$ Root an. cond. $8.47032$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.59·3-s + 20.9·5-s − 13.9·7-s + 4.32·9-s − 58.3·11-s − 65.0·13-s + 117.·15-s + 31.7·17-s − 19·19-s − 77.8·21-s − 151.·23-s + 314.·25-s − 126.·27-s − 110.·29-s − 94.0·31-s − 326.·33-s − 291.·35-s + 291.·37-s − 364.·39-s + 64.4·41-s + 449.·43-s + 90.7·45-s − 530.·47-s − 149.·49-s + 177.·51-s − 621.·53-s − 1.22e3·55-s + ⋯
 L(s)  = 1 + 1.07·3-s + 1.87·5-s − 0.750·7-s + 0.160·9-s − 1.59·11-s − 1.38·13-s + 2.01·15-s + 0.452·17-s − 0.229·19-s − 0.808·21-s − 1.37·23-s + 2.51·25-s − 0.904·27-s − 0.710·29-s − 0.544·31-s − 1.72·33-s − 1.40·35-s + 1.29·37-s − 1.49·39-s + 0.245·41-s + 1.59·43-s + 0.300·45-s − 1.64·47-s − 0.436·49-s + 0.487·51-s − 1.60·53-s − 2.99·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $-1$ Analytic conductor: $$71.7463$$ Root analytic conductor: $$8.47032$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1216,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + 19T$$
good3 $$1 - 5.59T + 27T^{2}$$
5 $$1 - 20.9T + 125T^{2}$$
7 $$1 + 13.9T + 343T^{2}$$
11 $$1 + 58.3T + 1.33e3T^{2}$$
13 $$1 + 65.0T + 2.19e3T^{2}$$
17 $$1 - 31.7T + 4.91e3T^{2}$$
23 $$1 + 151.T + 1.21e4T^{2}$$
29 $$1 + 110.T + 2.43e4T^{2}$$
31 $$1 + 94.0T + 2.97e4T^{2}$$
37 $$1 - 291.T + 5.06e4T^{2}$$
41 $$1 - 64.4T + 6.89e4T^{2}$$
43 $$1 - 449.T + 7.95e4T^{2}$$
47 $$1 + 530.T + 1.03e5T^{2}$$
53 $$1 + 621.T + 1.48e5T^{2}$$
59 $$1 + 244.T + 2.05e5T^{2}$$
61 $$1 + 801.T + 2.26e5T^{2}$$
67 $$1 + 7.34T + 3.00e5T^{2}$$
71 $$1 - 1.02e3T + 3.57e5T^{2}$$
73 $$1 - 592.T + 3.89e5T^{2}$$
79 $$1 + 120.T + 4.93e5T^{2}$$
83 $$1 + 502.T + 5.71e5T^{2}$$
89 $$1 - 253.T + 7.04e5T^{2}$$
97 $$1 - 511.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$