# 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 + 4.12·2-s − 254.·3-s − 494.·4-s + 151.·5-s − 1.04e3·6-s + 9.40e3·7-s − 4.15e3·8-s + 4.48e4·9-s + 625.·10-s − 5.69e4·11-s + 1.25e5·12-s − 6.08e4·13-s + 3.88e4·14-s − 3.85e4·15-s + 2.36e5·16-s + 8.35e4·17-s + 1.85e5·18-s + 1.00e6·19-s − 7.50e4·20-s − 2.39e6·21-s − 2.35e5·22-s + 1.35e6·23-s + 1.05e6·24-s − 1.93e6·25-s − 2.51e5·26-s − 6.39e6·27-s − 4.65e6·28-s + ⋯
 L(s)  = 1 + 0.182·2-s − 1.81·3-s − 0.966·4-s + 0.108·5-s − 0.330·6-s + 1.48·7-s − 0.358·8-s + 2.27·9-s + 0.0197·10-s − 1.17·11-s + 1.75·12-s − 0.591·13-s + 0.270·14-s − 0.196·15-s + 0.901·16-s + 0.242·17-s + 0.416·18-s + 1.77·19-s − 0.104·20-s − 2.68·21-s − 0.214·22-s + 1.01·23-s + 0.650·24-s − 0.988·25-s − 0.107·26-s − 2.31·27-s − 1.43·28-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

 $$L(5)$$ $$\approx$$ $$0.842276$$ $$L(\frac12)$$ $$\approx$$ $$0.842276$$ $$L(\frac{11}{2})$$ 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 - 4.12T + 512T^{2}$$
3 $$1 + 254.T + 1.96e4T^{2}$$
5 $$1 - 151.T + 1.95e6T^{2}$$
7 $$1 - 9.40e3T + 4.03e7T^{2}$$
11 $$1 + 5.69e4T + 2.35e9T^{2}$$
13 $$1 + 6.08e4T + 1.06e10T^{2}$$
19 $$1 - 1.00e6T + 3.22e11T^{2}$$
23 $$1 - 1.35e6T + 1.80e12T^{2}$$
29 $$1 - 3.12e6T + 1.45e13T^{2}$$
31 $$1 - 2.97e6T + 2.64e13T^{2}$$
37 $$1 - 6.81e5T + 1.29e14T^{2}$$
41 $$1 + 4.09e6T + 3.27e14T^{2}$$
43 $$1 - 1.00e7T + 5.02e14T^{2}$$
47 $$1 - 2.54e7T + 1.11e15T^{2}$$
53 $$1 + 3.14e7T + 3.29e15T^{2}$$
59 $$1 + 9.03e7T + 8.66e15T^{2}$$
61 $$1 - 9.87e7T + 1.16e16T^{2}$$
67 $$1 - 1.32e8T + 2.72e16T^{2}$$
71 $$1 - 4.18e8T + 4.58e16T^{2}$$
73 $$1 - 4.80e7T + 5.88e16T^{2}$$
79 $$1 - 3.49e7T + 1.19e17T^{2}$$
83 $$1 + 2.05e8T + 1.86e17T^{2}$$
89 $$1 + 2.03e8T + 3.50e17T^{2}$$
97 $$1 - 1.24e9T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$