# Properties

 Label 2-17e2-1.1-c3-0-53 Degree $2$ Conductor $289$ Sign $-1$ Analytic cond. $17.0515$ Root an. cond. $4.12935$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.227·2-s + 8.23·3-s − 7.94·4-s + 2.75·5-s + 1.87·6-s − 21.5·7-s − 3.63·8-s + 40.8·9-s + 0.626·10-s − 54.5·11-s − 65.4·12-s − 52.4·13-s − 4.90·14-s + 22.6·15-s + 62.7·16-s + 9.31·18-s − 19.6·19-s − 21.8·20-s − 177.·21-s − 12.4·22-s − 13.9·23-s − 29.9·24-s − 117.·25-s − 11.9·26-s + 114.·27-s + 171.·28-s − 70.0·29-s + ⋯
 L(s)  = 1 + 0.0805·2-s + 1.58·3-s − 0.993·4-s + 0.246·5-s + 0.127·6-s − 1.16·7-s − 0.160·8-s + 1.51·9-s + 0.0198·10-s − 1.49·11-s − 1.57·12-s − 1.11·13-s − 0.0936·14-s + 0.390·15-s + 0.980·16-s + 0.121·18-s − 0.237·19-s − 0.244·20-s − 1.84·21-s − 0.120·22-s − 0.126·23-s − 0.254·24-s − 0.939·25-s − 0.0902·26-s + 0.815·27-s + 1.15·28-s − 0.448·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$289$$    =    $$17^{2}$$ Sign: $-1$ Analytic conductor: $$17.0515$$ Root analytic conductor: $$4.12935$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 289,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad17 $$1$$
good2 $$1 - 0.227T + 8T^{2}$$
3 $$1 - 8.23T + 27T^{2}$$
5 $$1 - 2.75T + 125T^{2}$$
7 $$1 + 21.5T + 343T^{2}$$
11 $$1 + 54.5T + 1.33e3T^{2}$$
13 $$1 + 52.4T + 2.19e3T^{2}$$
19 $$1 + 19.6T + 6.85e3T^{2}$$
23 $$1 + 13.9T + 1.21e4T^{2}$$
29 $$1 + 70.0T + 2.43e4T^{2}$$
31 $$1 - 167.T + 2.97e4T^{2}$$
37 $$1 + 198.T + 5.06e4T^{2}$$
41 $$1 - 434.T + 6.89e4T^{2}$$
43 $$1 + 127.T + 7.95e4T^{2}$$
47 $$1 - 207.T + 1.03e5T^{2}$$
53 $$1 - 312.T + 1.48e5T^{2}$$
59 $$1 + 576.T + 2.05e5T^{2}$$
61 $$1 + 78.3T + 2.26e5T^{2}$$
67 $$1 + 359.T + 3.00e5T^{2}$$
71 $$1 - 213.T + 3.57e5T^{2}$$
73 $$1 - 29.0T + 3.89e5T^{2}$$
79 $$1 + 855.T + 4.93e5T^{2}$$
83 $$1 - 13.6T + 5.71e5T^{2}$$
89 $$1 + 651.T + 7.04e5T^{2}$$
97 $$1 - 1.19e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$