# 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 − 1.51·2-s + 5.95·3-s − 5.69·4-s − 9.03·6-s + 3.35·7-s + 20.7·8-s + 8.40·9-s + 20.2·11-s − 33.8·12-s + 5.57·13-s − 5.09·14-s + 13.9·16-s + 26.5·17-s − 12.7·18-s − 25.3·19-s + 19.9·21-s − 30.7·22-s + 90.2·23-s + 123.·24-s − 8.46·26-s − 110.·27-s − 19.1·28-s − 123.·29-s + 129.·31-s − 187.·32-s + 120.·33-s − 40.2·34-s + ⋯
 L(s)  = 1 − 0.536·2-s + 1.14·3-s − 0.711·4-s − 0.614·6-s + 0.181·7-s + 0.919·8-s + 0.311·9-s + 0.554·11-s − 0.814·12-s + 0.118·13-s − 0.0973·14-s + 0.218·16-s + 0.378·17-s − 0.167·18-s − 0.306·19-s + 0.207·21-s − 0.297·22-s + 0.818·23-s + 1.05·24-s − 0.0638·26-s − 0.788·27-s − 0.128·28-s − 0.789·29-s + 0.750·31-s − 1.03·32-s + 0.634·33-s − 0.203·34-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 = $(2,\ 1175,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $2.082544401$ $L(\frac12)$ $\approx$ $2.082544401$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{5,\;47\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{5,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1$$
47 $$1 + 47T$$
good2 $$1 + 1.51T + 8T^{2}$$
3 $$1 - 5.95T + 27T^{2}$$
7 $$1 - 3.35T + 343T^{2}$$
11 $$1 - 20.2T + 1.33e3T^{2}$$
13 $$1 - 5.57T + 2.19e3T^{2}$$
17 $$1 - 26.5T + 4.91e3T^{2}$$
19 $$1 + 25.3T + 6.85e3T^{2}$$
23 $$1 - 90.2T + 1.21e4T^{2}$$
29 $$1 + 123.T + 2.43e4T^{2}$$
31 $$1 - 129.T + 2.97e4T^{2}$$
37 $$1 - 213.T + 5.06e4T^{2}$$
41 $$1 + 124.T + 6.89e4T^{2}$$
43 $$1 - 424.T + 7.95e4T^{2}$$
53 $$1 + 361.T + 1.48e5T^{2}$$
59 $$1 - 836.T + 2.05e5T^{2}$$
61 $$1 + 194.T + 2.26e5T^{2}$$
67 $$1 + 902.T + 3.00e5T^{2}$$
71 $$1 - 690.T + 3.57e5T^{2}$$
73 $$1 - 698.T + 3.89e5T^{2}$$
79 $$1 + 449.T + 4.93e5T^{2}$$
83 $$1 - 543.T + 5.71e5T^{2}$$
89 $$1 - 725.T + 7.04e5T^{2}$$
97 $$1 - 214.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}