# Properties

 Label 2-17e2-1.1-c3-0-8 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 $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.03·2-s − 8.47·3-s + 17.3·4-s − 0.885·5-s + 42.6·6-s − 3.81·7-s − 46.9·8-s + 44.8·9-s + 4.45·10-s + 52.3·11-s − 146.·12-s − 8.06·13-s + 19.2·14-s + 7.50·15-s + 97.5·16-s − 225.·18-s − 66.5·19-s − 15.3·20-s + 32.3·21-s − 263.·22-s − 180.·23-s + 397.·24-s − 124.·25-s + 40.5·26-s − 151.·27-s − 66.1·28-s + 41.2·29-s + ⋯
 L(s)  = 1 − 1.77·2-s − 1.63·3-s + 2.16·4-s − 0.0792·5-s + 2.90·6-s − 0.206·7-s − 2.07·8-s + 1.66·9-s + 0.140·10-s + 1.43·11-s − 3.53·12-s − 0.171·13-s + 0.366·14-s + 0.129·15-s + 1.52·16-s − 2.95·18-s − 0.803·19-s − 0.171·20-s + 0.336·21-s − 2.55·22-s − 1.63·23-s + 3.38·24-s − 0.993·25-s + 0.305·26-s − 1.07·27-s − 0.446·28-s + 0.264·29-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: $$0$$ Selberg data: $$(2,\ 289,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.2963821998$$ $$L(\frac12)$$ $$\approx$$ $$0.2963821998$$ $$L(\frac{5}{2})$$ 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 + 5.03T + 8T^{2}$$
3 $$1 + 8.47T + 27T^{2}$$
5 $$1 + 0.885T + 125T^{2}$$
7 $$1 + 3.81T + 343T^{2}$$
11 $$1 - 52.3T + 1.33e3T^{2}$$
13 $$1 + 8.06T + 2.19e3T^{2}$$
19 $$1 + 66.5T + 6.85e3T^{2}$$
23 $$1 + 180.T + 1.21e4T^{2}$$
29 $$1 - 41.2T + 2.43e4T^{2}$$
31 $$1 - 34.9T + 2.97e4T^{2}$$
37 $$1 + 130.T + 5.06e4T^{2}$$
41 $$1 - 17.9T + 6.89e4T^{2}$$
43 $$1 - 277.T + 7.95e4T^{2}$$
47 $$1 - 463.T + 1.03e5T^{2}$$
53 $$1 + 329.T + 1.48e5T^{2}$$
59 $$1 - 678.T + 2.05e5T^{2}$$
61 $$1 + 340.T + 2.26e5T^{2}$$
67 $$1 - 15.3T + 3.00e5T^{2}$$
71 $$1 - 670.T + 3.57e5T^{2}$$
73 $$1 + 193.T + 3.89e5T^{2}$$
79 $$1 + 1.08e3T + 4.93e5T^{2}$$
83 $$1 + 865.T + 5.71e5T^{2}$$
89 $$1 - 1.12e3T + 7.04e5T^{2}$$
97 $$1 - 379.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$