# Properties

 Label 2-1045-1.1-c5-0-159 Degree $2$ Conductor $1045$ Sign $1$ Analytic cond. $167.601$ Root an. cond. $12.9460$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 0.779·2-s + 22.1·3-s − 31.3·4-s + 25·5-s + 17.2·6-s + 247.·7-s − 49.4·8-s + 246.·9-s + 19.4·10-s + 121·11-s − 694.·12-s − 1.10e3·13-s + 192.·14-s + 553.·15-s + 966.·16-s − 978.·17-s + 192.·18-s + 361·19-s − 784.·20-s + 5.47e3·21-s + 94.2·22-s + 552.·23-s − 1.09e3·24-s + 625·25-s − 861.·26-s + 84.4·27-s − 7.76e3·28-s + ⋯
 L(s)  = 1 + 0.137·2-s + 1.41·3-s − 0.981·4-s + 0.447·5-s + 0.195·6-s + 1.90·7-s − 0.272·8-s + 1.01·9-s + 0.0616·10-s + 0.301·11-s − 1.39·12-s − 1.81·13-s + 0.262·14-s + 0.634·15-s + 0.943·16-s − 0.821·17-s + 0.139·18-s + 0.229·19-s − 0.438·20-s + 2.70·21-s + 0.0415·22-s + 0.217·23-s − 0.387·24-s + 0.200·25-s − 0.250·26-s + 0.0222·27-s − 1.87·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $1$ Analytic conductor: $$167.601$$ Root analytic conductor: $$12.9460$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1045,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$4.592418528$$ $$L(\frac12)$$ $$\approx$$ $$4.592418528$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - 25T$$
11 $$1 - 121T$$
19 $$1 - 361T$$
good2 $$1 - 0.779T + 32T^{2}$$
3 $$1 - 22.1T + 243T^{2}$$
7 $$1 - 247.T + 1.68e4T^{2}$$
13 $$1 + 1.10e3T + 3.71e5T^{2}$$
17 $$1 + 978.T + 1.41e6T^{2}$$
23 $$1 - 552.T + 6.43e6T^{2}$$
29 $$1 - 6.16e3T + 2.05e7T^{2}$$
31 $$1 - 8.83e3T + 2.86e7T^{2}$$
37 $$1 + 1.48e4T + 6.93e7T^{2}$$
41 $$1 + 1.53e4T + 1.15e8T^{2}$$
43 $$1 - 1.71e4T + 1.47e8T^{2}$$
47 $$1 - 2.14e4T + 2.29e8T^{2}$$
53 $$1 - 2.15e4T + 4.18e8T^{2}$$
59 $$1 - 3.29e4T + 7.14e8T^{2}$$
61 $$1 - 4.14e4T + 8.44e8T^{2}$$
67 $$1 + 1.42e4T + 1.35e9T^{2}$$
71 $$1 + 3.04e4T + 1.80e9T^{2}$$
73 $$1 + 1.99e4T + 2.07e9T^{2}$$
79 $$1 - 4.46e4T + 3.07e9T^{2}$$
83 $$1 - 1.13e5T + 3.93e9T^{2}$$
89 $$1 + 4.75e4T + 5.58e9T^{2}$$
97 $$1 + 8.69e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$