# Properties

 Degree $2$ Conductor $17$ Sign $1$ Motivic weight $9$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 16.8·2-s − 116.·3-s − 229.·4-s − 1.10e3·5-s + 1.96e3·6-s − 5.16e3·7-s + 1.24e4·8-s − 6.02e3·9-s + 1.85e4·10-s + 4.45e4·11-s + 2.68e4·12-s + 6.96e4·13-s + 8.68e4·14-s + 1.28e5·15-s − 9.20e4·16-s + 8.35e4·17-s + 1.01e5·18-s − 1.70e5·19-s + 2.53e5·20-s + 6.03e5·21-s − 7.48e5·22-s − 2.00e6·23-s − 1.45e6·24-s − 7.35e5·25-s − 1.17e6·26-s + 3.00e6·27-s + 1.18e6·28-s + ⋯
 L(s)  = 1 − 0.742·2-s − 0.833·3-s − 0.447·4-s − 0.789·5-s + 0.619·6-s − 0.812·7-s + 1.07·8-s − 0.305·9-s + 0.586·10-s + 0.917·11-s + 0.373·12-s + 0.676·13-s + 0.604·14-s + 0.657·15-s − 0.351·16-s + 0.242·17-s + 0.227·18-s − 0.299·19-s + 0.353·20-s + 0.677·21-s − 0.681·22-s − 1.49·23-s − 0.896·24-s − 0.376·25-s − 0.502·26-s + 1.08·27-s + 0.364·28-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: $$0$$ Selberg data: $$(2,\ 17,\ (\ :9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$0.423081$$ $$L(\frac12)$$ $$\approx$$ $$0.423081$$ $$L(\frac{11}{2})$$ 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 + 16.8T + 512T^{2}$$
3 $$1 + 116.T + 1.96e4T^{2}$$
5 $$1 + 1.10e3T + 1.95e6T^{2}$$
7 $$1 + 5.16e3T + 4.03e7T^{2}$$
11 $$1 - 4.45e4T + 2.35e9T^{2}$$
13 $$1 - 6.96e4T + 1.06e10T^{2}$$
19 $$1 + 1.70e5T + 3.22e11T^{2}$$
23 $$1 + 2.00e6T + 1.80e12T^{2}$$
29 $$1 - 1.55e5T + 1.45e13T^{2}$$
31 $$1 - 4.27e6T + 2.64e13T^{2}$$
37 $$1 - 1.51e7T + 1.29e14T^{2}$$
41 $$1 - 1.59e7T + 3.27e14T^{2}$$
43 $$1 - 1.49e7T + 5.02e14T^{2}$$
47 $$1 - 3.36e7T + 1.11e15T^{2}$$
53 $$1 + 5.50e7T + 3.29e15T^{2}$$
59 $$1 + 7.94e7T + 8.66e15T^{2}$$
61 $$1 - 1.27e7T + 1.16e16T^{2}$$
67 $$1 - 2.73e8T + 2.72e16T^{2}$$
71 $$1 - 3.88e6T + 4.58e16T^{2}$$
73 $$1 - 2.32e8T + 5.88e16T^{2}$$
79 $$1 - 3.38e8T + 1.19e17T^{2}$$
83 $$1 - 5.74e8T + 1.86e17T^{2}$$
89 $$1 + 9.82e8T + 3.50e17T^{2}$$
97 $$1 + 1.03e9T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$