# Properties

 Degree $2$ Conductor $17$ Sign $-1$ Motivic weight $9$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 28.1·2-s − 3.02·3-s + 281.·4-s + 762.·5-s + 85.1·6-s + 5.57e3·7-s + 6.49e3·8-s − 1.96e4·9-s − 2.14e4·10-s − 4.76e4·11-s − 850.·12-s − 9.22e4·13-s − 1.56e5·14-s − 2.30e3·15-s − 3.27e5·16-s − 8.35e4·17-s + 5.54e5·18-s − 8.37e3·19-s + 2.14e5·20-s − 1.68e4·21-s + 1.34e6·22-s + 3.64e5·23-s − 1.96e4·24-s − 1.37e6·25-s + 2.59e6·26-s + 1.19e5·27-s + 1.56e6·28-s + ⋯
 L(s)  = 1 − 1.24·2-s − 0.0215·3-s + 0.549·4-s + 0.545·5-s + 0.0268·6-s + 0.877·7-s + 0.560·8-s − 0.999·9-s − 0.679·10-s − 0.981·11-s − 0.0118·12-s − 0.895·13-s − 1.09·14-s − 0.0117·15-s − 1.24·16-s − 0.242·17-s + 1.24·18-s − 0.0147·19-s + 0.299·20-s − 0.0189·21-s + 1.22·22-s + 0.271·23-s − 0.0120·24-s − 0.702·25-s + 1.11·26-s + 0.0430·27-s + 0.481·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$17$$ Sign: $-1$ Motivic weight: $$9$$ Character: $\chi_{17} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 17,\ (\ :9/2),\ -1)$$

## Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad17 $$1 + 8.35e4T$$
good2 $$1 + 28.1T + 512T^{2}$$
3 $$1 + 3.02T + 1.96e4T^{2}$$
5 $$1 - 762.T + 1.95e6T^{2}$$
7 $$1 - 5.57e3T + 4.03e7T^{2}$$
11 $$1 + 4.76e4T + 2.35e9T^{2}$$
13 $$1 + 9.22e4T + 1.06e10T^{2}$$
19 $$1 + 8.37e3T + 3.22e11T^{2}$$
23 $$1 - 3.64e5T + 1.80e12T^{2}$$
29 $$1 + 3.50e6T + 1.45e13T^{2}$$
31 $$1 + 5.20e6T + 2.64e13T^{2}$$
37 $$1 + 4.99e5T + 1.29e14T^{2}$$
41 $$1 + 5.43e6T + 3.27e14T^{2}$$
43 $$1 + 3.54e7T + 5.02e14T^{2}$$
47 $$1 - 1.21e7T + 1.11e15T^{2}$$
53 $$1 - 1.04e8T + 3.29e15T^{2}$$
59 $$1 - 4.16e7T + 8.66e15T^{2}$$
61 $$1 - 5.67e7T + 1.16e16T^{2}$$
67 $$1 + 1.74e8T + 2.72e16T^{2}$$
71 $$1 - 3.46e7T + 4.58e16T^{2}$$
73 $$1 + 3.93e8T + 5.88e16T^{2}$$
79 $$1 - 1.85e8T + 1.19e17T^{2}$$
83 $$1 - 3.62e8T + 1.86e17T^{2}$$
89 $$1 + 5.04e7T + 3.50e17T^{2}$$
97 $$1 + 9.67e8T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$