# Properties

 Degree 2 Conductor 17 Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.67·2-s − 7.62·3-s + 13.8·4-s − 11.9·5-s − 35.6·6-s + 26.1·7-s + 27.1·8-s + 31.2·9-s − 55.6·10-s − 3.24·11-s − 105.·12-s − 20.0·13-s + 122.·14-s + 90.9·15-s + 16.4·16-s − 17·17-s + 145.·18-s + 57.3·19-s − 164.·20-s − 199.·21-s − 15.1·22-s + 77.0·23-s − 207.·24-s + 17.0·25-s − 93.6·26-s − 32.1·27-s + 361.·28-s + ⋯
 L(s)  = 1 + 1.65·2-s − 1.46·3-s + 1.72·4-s − 1.06·5-s − 2.42·6-s + 1.41·7-s + 1.20·8-s + 1.15·9-s − 1.76·10-s − 0.0889·11-s − 2.53·12-s − 0.427·13-s + 2.32·14-s + 1.56·15-s + 0.257·16-s − 0.242·17-s + 1.90·18-s + 0.692·19-s − 1.84·20-s − 2.07·21-s − 0.146·22-s + 0.698·23-s − 1.76·24-s + 0.136·25-s − 0.706·26-s − 0.229·27-s + 2.43·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$17$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{17} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 17,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $1.47255$ $L(\frac12)$ $\approx$ $1.47255$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 17$, $$F_p$$ is a polynomial of degree 2. If $p = 17$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad17 $$1 + 17T$$
good2 $$1 - 4.67T + 8T^{2}$$
3 $$1 + 7.62T + 27T^{2}$$
5 $$1 + 11.9T + 125T^{2}$$
7 $$1 - 26.1T + 343T^{2}$$
11 $$1 + 3.24T + 1.33e3T^{2}$$
13 $$1 + 20.0T + 2.19e3T^{2}$$
19 $$1 - 57.3T + 6.85e3T^{2}$$
23 $$1 - 77.0T + 1.21e4T^{2}$$
29 $$1 + 286.T + 2.43e4T^{2}$$
31 $$1 + 8.54T + 2.97e4T^{2}$$
37 $$1 - 357.T + 5.06e4T^{2}$$
41 $$1 - 194.T + 6.89e4T^{2}$$
43 $$1 + 74.2T + 7.95e4T^{2}$$
47 $$1 - 23.6T + 1.03e5T^{2}$$
53 $$1 - 104.T + 1.48e5T^{2}$$
59 $$1 - 249.T + 2.05e5T^{2}$$
61 $$1 + 370.T + 2.26e5T^{2}$$
67 $$1 - 939.T + 3.00e5T^{2}$$
71 $$1 + 520.T + 3.57e5T^{2}$$
73 $$1 - 348.T + 3.89e5T^{2}$$
79 $$1 + 953.T + 4.93e5T^{2}$$
83 $$1 + 1.41e3T + 5.71e5T^{2}$$
89 $$1 + 486.T + 7.04e5T^{2}$$
97 $$1 + 685.T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}