# 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 − 1.22·2-s + 177.·3-s − 510.·4-s − 1.62e3·5-s − 217.·6-s − 1.83e3·7-s + 1.25e3·8-s + 1.18e4·9-s + 1.98e3·10-s − 3.17e4·11-s − 9.05e4·12-s − 1.32e5·13-s + 2.25e3·14-s − 2.87e5·15-s + 2.59e5·16-s − 8.35e4·17-s − 1.44e4·18-s − 1.60e3·19-s + 8.27e5·20-s − 3.25e5·21-s + 3.90e4·22-s + 2.32e4·23-s + 2.22e5·24-s + 6.71e5·25-s + 1.62e5·26-s − 1.39e6·27-s + 9.36e5·28-s + ⋯
 L(s)  = 1 − 0.0542·2-s + 1.26·3-s − 0.997·4-s − 1.15·5-s − 0.0685·6-s − 0.288·7-s + 0.108·8-s + 0.599·9-s + 0.0628·10-s − 0.654·11-s − 1.26·12-s − 1.28·13-s + 0.0156·14-s − 1.46·15-s + 0.991·16-s − 0.242·17-s − 0.0325·18-s − 0.00282·19-s + 1.15·20-s − 0.365·21-s + 0.0354·22-s + 0.0173·23-s + 0.136·24-s + 0.343·25-s + 0.0697·26-s − 0.506·27-s + 0.287·28-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 + 1.22T + 512T^{2}$$
3 $$1 - 177.T + 1.96e4T^{2}$$
5 $$1 + 1.62e3T + 1.95e6T^{2}$$
7 $$1 + 1.83e3T + 4.03e7T^{2}$$
11 $$1 + 3.17e4T + 2.35e9T^{2}$$
13 $$1 + 1.32e5T + 1.06e10T^{2}$$
19 $$1 + 1.60e3T + 3.22e11T^{2}$$
23 $$1 - 2.32e4T + 1.80e12T^{2}$$
29 $$1 - 3.73e6T + 1.45e13T^{2}$$
31 $$1 - 8.91e6T + 2.64e13T^{2}$$
37 $$1 + 1.20e7T + 1.29e14T^{2}$$
41 $$1 + 1.26e7T + 3.27e14T^{2}$$
43 $$1 - 2.86e7T + 5.02e14T^{2}$$
47 $$1 + 7.17e6T + 1.11e15T^{2}$$
53 $$1 + 5.96e7T + 3.29e15T^{2}$$
59 $$1 - 1.85e8T + 8.66e15T^{2}$$
61 $$1 + 2.00e8T + 1.16e16T^{2}$$
67 $$1 + 1.27e8T + 2.72e16T^{2}$$
71 $$1 + 3.27e8T + 4.58e16T^{2}$$
73 $$1 + 1.48e8T + 5.88e16T^{2}$$
79 $$1 + 2.58e8T + 1.19e17T^{2}$$
83 $$1 - 3.45e8T + 1.86e17T^{2}$$
89 $$1 - 4.03e8T + 3.50e17T^{2}$$
97 $$1 + 9.89e8T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$