# 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 + 4.90·2-s + 0.777·3-s + 16.1·4-s + 3.81·6-s + 30.3·7-s + 39.7·8-s − 26.3·9-s + 22.4·11-s + 12.5·12-s + 62.0·13-s + 148.·14-s + 66.4·16-s + 72.1·17-s − 129.·18-s − 25.0·19-s + 23.5·21-s + 110.·22-s − 103.·23-s + 30.9·24-s + 304.·26-s − 41.5·27-s + 488.·28-s − 234.·29-s + 198.·31-s + 8.17·32-s + 17.4·33-s + 354.·34-s + ⋯
 L(s)  = 1 + 1.73·2-s + 0.149·3-s + 2.01·4-s + 0.259·6-s + 1.63·7-s + 1.75·8-s − 0.977·9-s + 0.614·11-s + 0.301·12-s + 1.32·13-s + 2.84·14-s + 1.03·16-s + 1.02·17-s − 1.69·18-s − 0.302·19-s + 0.245·21-s + 1.06·22-s − 0.935·23-s + 0.263·24-s + 2.29·26-s − 0.296·27-s + 3.29·28-s − 1.50·29-s + 1.15·31-s + 0.0451·32-s + 0.0920·33-s + 1.78·34-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 = $(2,\ 1175,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $8.455390720$ $L(\frac12)$ $\approx$ $8.455390720$ $L(\frac{5}{2})$ 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 - 4.90T + 8T^{2}$$
3 $$1 - 0.777T + 27T^{2}$$
7 $$1 - 30.3T + 343T^{2}$$
11 $$1 - 22.4T + 1.33e3T^{2}$$
13 $$1 - 62.0T + 2.19e3T^{2}$$
17 $$1 - 72.1T + 4.91e3T^{2}$$
19 $$1 + 25.0T + 6.85e3T^{2}$$
23 $$1 + 103.T + 1.21e4T^{2}$$
29 $$1 + 234.T + 2.43e4T^{2}$$
31 $$1 - 198.T + 2.97e4T^{2}$$
37 $$1 - 203.T + 5.06e4T^{2}$$
41 $$1 - 210.T + 6.89e4T^{2}$$
43 $$1 + 111.T + 7.95e4T^{2}$$
53 $$1 + 499.T + 1.48e5T^{2}$$
59 $$1 + 562.T + 2.05e5T^{2}$$
61 $$1 - 548.T + 2.26e5T^{2}$$
67 $$1 - 760.T + 3.00e5T^{2}$$
71 $$1 + 668.T + 3.57e5T^{2}$$
73 $$1 - 1.14e3T + 3.89e5T^{2}$$
79 $$1 - 975.T + 4.93e5T^{2}$$
83 $$1 + 698.T + 5.71e5T^{2}$$
89 $$1 - 451.T + 7.04e5T^{2}$$
97 $$1 - 390.T + 9.12e5T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}