# Properties

 Degree 2 Conductor $5^{2} \cdot 47$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.60·2-s − 1.72·3-s − 5.41·4-s − 2.78·6-s + 11.3·7-s − 21.5·8-s − 24.0·9-s − 40.6·11-s + 9.35·12-s + 12.3·13-s + 18.2·14-s + 8.55·16-s − 59.6·17-s − 38.6·18-s + 26.4·19-s − 19.5·21-s − 65.4·22-s − 107.·23-s + 37.2·24-s + 19.9·26-s + 88.1·27-s − 61.2·28-s + 173.·29-s − 332.·31-s + 186.·32-s + 70.2·33-s − 96.0·34-s + ⋯
 L(s)  = 1 + 0.568·2-s − 0.332·3-s − 0.676·4-s − 0.189·6-s + 0.611·7-s − 0.953·8-s − 0.889·9-s − 1.11·11-s + 0.224·12-s + 0.264·13-s + 0.347·14-s + 0.133·16-s − 0.851·17-s − 0.506·18-s + 0.318·19-s − 0.203·21-s − 0.633·22-s − 0.970·23-s + 0.317·24-s + 0.150·26-s + 0.628·27-s − 0.413·28-s + 1.11·29-s − 1.92·31-s + 1.02·32-s + 0.370·33-s − 0.484·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1175$$    =    $$5^{2} \cdot 47$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{1175} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 1175,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $1.077468228$ $L(\frac12)$ $\approx$ $1.077468228$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;47\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{5,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1$$
47 $$1 + 47T$$
good2 $$1 - 1.60T + 8T^{2}$$
3 $$1 + 1.72T + 27T^{2}$$
7 $$1 - 11.3T + 343T^{2}$$
11 $$1 + 40.6T + 1.33e3T^{2}$$
13 $$1 - 12.3T + 2.19e3T^{2}$$
17 $$1 + 59.6T + 4.91e3T^{2}$$
19 $$1 - 26.4T + 6.85e3T^{2}$$
23 $$1 + 107.T + 1.21e4T^{2}$$
29 $$1 - 173.T + 2.43e4T^{2}$$
31 $$1 + 332.T + 2.97e4T^{2}$$
37 $$1 - 172.T + 5.06e4T^{2}$$
41 $$1 + 178.T + 6.89e4T^{2}$$
43 $$1 + 63.2T + 7.95e4T^{2}$$
53 $$1 - 402.T + 1.48e5T^{2}$$
59 $$1 - 305.T + 2.05e5T^{2}$$
61 $$1 + 86.2T + 2.26e5T^{2}$$
67 $$1 - 681.T + 3.00e5T^{2}$$
71 $$1 - 726.T + 3.57e5T^{2}$$
73 $$1 - 79.8T + 3.89e5T^{2}$$
79 $$1 - 279.T + 4.93e5T^{2}$$
83 $$1 + 556.T + 5.71e5T^{2}$$
89 $$1 + 342.T + 7.04e5T^{2}$$
97 $$1 - 1.63e3T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}