# Properties

 Label 2-17e2-1.1-c3-0-57 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 + 4.28·2-s − 2.85·3-s + 10.3·4-s + 6.36·5-s − 12.2·6-s − 29.4·7-s + 10.2·8-s − 18.8·9-s + 27.3·10-s − 61.3·11-s − 29.6·12-s + 18.5·13-s − 126.·14-s − 18.1·15-s − 39.1·16-s − 80.9·18-s + 115.·19-s + 66.1·20-s + 83.8·21-s − 263.·22-s + 7.38·23-s − 29.2·24-s − 84.4·25-s + 79.5·26-s + 130.·27-s − 305.·28-s − 164.·29-s + ⋯
 L(s)  = 1 + 1.51·2-s − 0.548·3-s + 1.29·4-s + 0.569·5-s − 0.831·6-s − 1.58·7-s + 0.453·8-s − 0.699·9-s + 0.863·10-s − 1.68·11-s − 0.712·12-s + 0.395·13-s − 2.40·14-s − 0.312·15-s − 0.611·16-s − 1.05·18-s + 1.39·19-s + 0.739·20-s + 0.871·21-s − 2.55·22-s + 0.0669·23-s − 0.248·24-s − 0.675·25-s + 0.600·26-s + 0.932·27-s − 2.06·28-s − 1.05·29-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 - 4.28T + 8T^{2}$$
3 $$1 + 2.85T + 27T^{2}$$
5 $$1 - 6.36T + 125T^{2}$$
7 $$1 + 29.4T + 343T^{2}$$
11 $$1 + 61.3T + 1.33e3T^{2}$$
13 $$1 - 18.5T + 2.19e3T^{2}$$
19 $$1 - 115.T + 6.85e3T^{2}$$
23 $$1 - 7.38T + 1.21e4T^{2}$$
29 $$1 + 164.T + 2.43e4T^{2}$$
31 $$1 - 127.T + 2.97e4T^{2}$$
37 $$1 - 158.T + 5.06e4T^{2}$$
41 $$1 - 31.3T + 6.89e4T^{2}$$
43 $$1 - 157.T + 7.95e4T^{2}$$
47 $$1 + 460.T + 1.03e5T^{2}$$
53 $$1 + 166.T + 1.48e5T^{2}$$
59 $$1 + 343.T + 2.05e5T^{2}$$
61 $$1 + 112.T + 2.26e5T^{2}$$
67 $$1 + 984.T + 3.00e5T^{2}$$
71 $$1 + 524.T + 3.57e5T^{2}$$
73 $$1 - 852.T + 3.89e5T^{2}$$
79 $$1 + 201.T + 4.93e5T^{2}$$
83 $$1 + 22.5T + 5.71e5T^{2}$$
89 $$1 - 502.T + 7.04e5T^{2}$$
97 $$1 - 680.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$