# Properties

 Label 2-17e2-1.1-c3-0-29 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 + 0.447·2-s − 9.51·3-s − 7.79·4-s + 11.7·5-s − 4.25·6-s − 2.27·7-s − 7.07·8-s + 63.5·9-s + 5.25·10-s − 22.6·11-s + 74.2·12-s + 67.4·13-s − 1.01·14-s − 111.·15-s + 59.2·16-s + 28.4·18-s + 42.9·19-s − 91.5·20-s + 21.6·21-s − 10.1·22-s − 117.·23-s + 67.2·24-s + 12.8·25-s + 30.1·26-s − 347.·27-s + 17.7·28-s − 226.·29-s + ⋯
 L(s)  = 1 + 0.158·2-s − 1.83·3-s − 0.974·4-s + 1.05·5-s − 0.289·6-s − 0.122·7-s − 0.312·8-s + 2.35·9-s + 0.166·10-s − 0.621·11-s + 1.78·12-s + 1.43·13-s − 0.0194·14-s − 1.92·15-s + 0.925·16-s + 0.372·18-s + 0.519·19-s − 1.02·20-s + 0.225·21-s − 0.0984·22-s − 1.06·23-s + 0.572·24-s + 0.103·25-s + 0.227·26-s − 2.47·27-s + 0.119·28-s − 1.44·29-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 - 0.447T + 8T^{2}$$
3 $$1 + 9.51T + 27T^{2}$$
5 $$1 - 11.7T + 125T^{2}$$
7 $$1 + 2.27T + 343T^{2}$$
11 $$1 + 22.6T + 1.33e3T^{2}$$
13 $$1 - 67.4T + 2.19e3T^{2}$$
19 $$1 - 42.9T + 6.85e3T^{2}$$
23 $$1 + 117.T + 1.21e4T^{2}$$
29 $$1 + 226.T + 2.43e4T^{2}$$
31 $$1 - 0.673T + 2.97e4T^{2}$$
37 $$1 - 99.9T + 5.06e4T^{2}$$
41 $$1 - 154.T + 6.89e4T^{2}$$
43 $$1 + 321.T + 7.95e4T^{2}$$
47 $$1 + 30.4T + 1.03e5T^{2}$$
53 $$1 + 361.T + 1.48e5T^{2}$$
59 $$1 - 147.T + 2.05e5T^{2}$$
61 $$1 + 321.T + 2.26e5T^{2}$$
67 $$1 - 612.T + 3.00e5T^{2}$$
71 $$1 + 248.T + 3.57e5T^{2}$$
73 $$1 - 701.T + 3.89e5T^{2}$$
79 $$1 + 773.T + 4.93e5T^{2}$$
83 $$1 + 1.00e3T + 5.71e5T^{2}$$
89 $$1 - 1.64e3T + 7.04e5T^{2}$$
97 $$1 - 479.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$