# 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 − 42.0·2-s − 256.·3-s + 1.25e3·4-s + 1.40e3·5-s + 1.08e4·6-s − 5.13e3·7-s − 3.13e4·8-s + 4.62e4·9-s − 5.91e4·10-s + 2.65e4·11-s − 3.22e5·12-s + 7.14e4·13-s + 2.16e5·14-s − 3.61e5·15-s + 6.74e5·16-s − 8.35e4·17-s − 1.94e6·18-s − 5.48e5·19-s + 1.76e6·20-s + 1.31e6·21-s − 1.11e6·22-s + 1.15e6·23-s + 8.04e6·24-s + 2.72e4·25-s − 3.00e6·26-s − 6.82e6·27-s − 6.46e6·28-s + ⋯
 L(s)  = 1 − 1.85·2-s − 1.83·3-s + 2.45·4-s + 1.00·5-s + 3.40·6-s − 0.808·7-s − 2.70·8-s + 2.34·9-s − 1.87·10-s + 0.546·11-s − 4.49·12-s + 0.693·13-s + 1.50·14-s − 1.84·15-s + 2.57·16-s − 0.242·17-s − 4.36·18-s − 0.965·19-s + 2.47·20-s + 1.48·21-s − 1.01·22-s + 0.864·23-s + 4.95·24-s + 0.0139·25-s − 1.28·26-s − 2.47·27-s − 1.98·28-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 + 42.0T + 512T^{2}$$
3 $$1 + 256.T + 1.96e4T^{2}$$
5 $$1 - 1.40e3T + 1.95e6T^{2}$$
7 $$1 + 5.13e3T + 4.03e7T^{2}$$
11 $$1 - 2.65e4T + 2.35e9T^{2}$$
13 $$1 - 7.14e4T + 1.06e10T^{2}$$
19 $$1 + 5.48e5T + 3.22e11T^{2}$$
23 $$1 - 1.15e6T + 1.80e12T^{2}$$
29 $$1 - 1.44e6T + 1.45e13T^{2}$$
31 $$1 + 6.05e6T + 2.64e13T^{2}$$
37 $$1 + 9.50e6T + 1.29e14T^{2}$$
41 $$1 - 1.75e7T + 3.27e14T^{2}$$
43 $$1 + 2.06e7T + 5.02e14T^{2}$$
47 $$1 - 3.15e7T + 1.11e15T^{2}$$
53 $$1 + 1.02e8T + 3.29e15T^{2}$$
59 $$1 + 5.95e7T + 8.66e15T^{2}$$
61 $$1 - 5.34e7T + 1.16e16T^{2}$$
67 $$1 + 6.45e7T + 2.72e16T^{2}$$
71 $$1 + 2.71e8T + 4.58e16T^{2}$$
73 $$1 - 2.75e8T + 5.88e16T^{2}$$
79 $$1 + 4.33e8T + 1.19e17T^{2}$$
83 $$1 - 1.23e8T + 1.86e17T^{2}$$
89 $$1 + 8.87e8T + 3.50e17T^{2}$$
97 $$1 + 4.41e8T + 7.60e17T^{2}$$
show less
$$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

−16.74617914817934616029540958849, −15.74625118886963632112456694456, −12.72126114348821388618046082561, −11.23240595320376213511309813856, −10.34105374452628142326963725185, −9.231449938266936398061284756306, −6.80615706385180649153702521673, −6.01247830173674070548198063867, −1.46726334419730167440098000473, 0, 1.46726334419730167440098000473, 6.01247830173674070548198063867, 6.80615706385180649153702521673, 9.231449938266936398061284756306, 10.34105374452628142326963725185, 11.23240595320376213511309813856, 12.72126114348821388618046082561, 15.74625118886963632112456694456, 16.74617914817934616029540958849