# Properties

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

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## Dirichlet series

 L(s)  = 1 + 43.1·2-s − 171.·3-s + 1.34e3·4-s + 1.53e3·5-s − 7.37e3·6-s + 3.02e3·7-s + 3.60e4·8-s + 9.56e3·9-s + 6.62e4·10-s + 5.19e4·11-s − 2.30e5·12-s − 1.66e5·13-s + 1.30e5·14-s − 2.62e5·15-s + 8.63e5·16-s + 8.35e4·17-s + 4.12e5·18-s − 1.03e6·19-s + 2.06e6·20-s − 5.17e5·21-s + 2.24e6·22-s − 6.47e5·23-s − 6.16e6·24-s + 4.06e5·25-s − 7.17e6·26-s + 1.73e6·27-s + 4.07e6·28-s + ⋯
 L(s)  = 1 + 1.90·2-s − 1.21·3-s + 2.63·4-s + 1.09·5-s − 2.32·6-s + 0.476·7-s + 3.10·8-s + 0.486·9-s + 2.09·10-s + 1.07·11-s − 3.20·12-s − 1.61·13-s + 0.908·14-s − 1.33·15-s + 3.29·16-s + 0.242·17-s + 0.926·18-s − 1.82·19-s + 2.89·20-s − 0.581·21-s + 2.03·22-s − 0.482·23-s − 3.79·24-s + 0.208·25-s − 3.07·26-s + 0.626·27-s + 1.25·28-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

 $$L(5)$$ $$\approx$$ $$4.17223$$ $$L(\frac12)$$ $$\approx$$ $$4.17223$$ $$L(\frac{11}{2})$$ 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 - 43.1T + 512T^{2}$$
3 $$1 + 171.T + 1.96e4T^{2}$$
5 $$1 - 1.53e3T + 1.95e6T^{2}$$
7 $$1 - 3.02e3T + 4.03e7T^{2}$$
11 $$1 - 5.19e4T + 2.35e9T^{2}$$
13 $$1 + 1.66e5T + 1.06e10T^{2}$$
19 $$1 + 1.03e6T + 3.22e11T^{2}$$
23 $$1 + 6.47e5T + 1.80e12T^{2}$$
29 $$1 - 1.01e5T + 1.45e13T^{2}$$
31 $$1 - 1.03e6T + 2.64e13T^{2}$$
37 $$1 + 5.58e6T + 1.29e14T^{2}$$
41 $$1 + 4.18e6T + 3.27e14T^{2}$$
43 $$1 + 9.60e6T + 5.02e14T^{2}$$
47 $$1 - 3.98e7T + 1.11e15T^{2}$$
53 $$1 - 6.22e7T + 3.29e15T^{2}$$
59 $$1 - 9.60e7T + 8.66e15T^{2}$$
61 $$1 + 1.86e8T + 1.16e16T^{2}$$
67 $$1 - 3.73e7T + 2.72e16T^{2}$$
71 $$1 - 2.04e8T + 4.58e16T^{2}$$
73 $$1 + 1.95e8T + 5.88e16T^{2}$$
79 $$1 - 2.72e8T + 1.19e17T^{2}$$
83 $$1 + 1.96e8T + 1.86e17T^{2}$$
89 $$1 + 3.93e8T + 3.50e17T^{2}$$
97 $$1 - 8.75e8T + 7.60e17T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−16.80148426133319821664368038389, −14.92849588036593211185287576832, −14.04680947809212863089612676546, −12.56521244844904325896824088305, −11.72703871362821385519538187084, −10.38762271031204590068908950068, −6.71926124848699301595070906182, −5.71504062958116055510467538005, −4.55855265695563168355632802467, −2.05266453409891427308902586184, 2.05266453409891427308902586184, 4.55855265695563168355632802467, 5.71504062958116055510467538005, 6.71926124848699301595070906182, 10.38762271031204590068908950068, 11.72703871362821385519538187084, 12.56521244844904325896824088305, 14.04680947809212863089612676546, 14.92849588036593211185287576832, 16.80148426133319821664368038389