# Properties

 Label 2-1045-1.1-c5-0-86 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 − 6.34·2-s − 22.4·3-s + 8.30·4-s + 25·5-s + 142.·6-s + 214.·7-s + 150.·8-s + 261.·9-s − 158.·10-s + 121·11-s − 186.·12-s − 442.·13-s − 1.36e3·14-s − 561.·15-s − 1.22e3·16-s − 1.31e3·17-s − 1.65e3·18-s + 361·19-s + 207.·20-s − 4.82e3·21-s − 768.·22-s + 2.15e3·23-s − 3.37e3·24-s + 625·25-s + 2.80e3·26-s − 408.·27-s + 1.78e3·28-s + ⋯
 L(s)  = 1 − 1.12·2-s − 1.44·3-s + 0.259·4-s + 0.447·5-s + 1.61·6-s + 1.65·7-s + 0.831·8-s + 1.07·9-s − 0.501·10-s + 0.301·11-s − 0.373·12-s − 0.726·13-s − 1.86·14-s − 0.644·15-s − 1.19·16-s − 1.10·17-s − 1.20·18-s + 0.229·19-s + 0.116·20-s − 2.38·21-s − 0.338·22-s + 0.849·23-s − 1.19·24-s + 0.200·25-s + 0.815·26-s − 0.107·27-s + 0.430·28-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

 $$L(3)$$ $$\approx$$ $$0.8846059162$$ $$L(\frac12)$$ $$\approx$$ $$0.8846059162$$ $$L(\frac{7}{2})$$ 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 + 6.34T + 32T^{2}$$
3 $$1 + 22.4T + 243T^{2}$$
7 $$1 - 214.T + 1.68e4T^{2}$$
13 $$1 + 442.T + 3.71e5T^{2}$$
17 $$1 + 1.31e3T + 1.41e6T^{2}$$
23 $$1 - 2.15e3T + 6.43e6T^{2}$$
29 $$1 - 5.03e3T + 2.05e7T^{2}$$
31 $$1 + 436.T + 2.86e7T^{2}$$
37 $$1 - 3.96e3T + 6.93e7T^{2}$$
41 $$1 - 8.18e3T + 1.15e8T^{2}$$
43 $$1 - 2.46e3T + 1.47e8T^{2}$$
47 $$1 - 1.71e4T + 2.29e8T^{2}$$
53 $$1 + 2.39e4T + 4.18e8T^{2}$$
59 $$1 + 3.13e4T + 7.14e8T^{2}$$
61 $$1 - 4.25e4T + 8.44e8T^{2}$$
67 $$1 - 3.53e3T + 1.35e9T^{2}$$
71 $$1 + 3.45e4T + 1.80e9T^{2}$$
73 $$1 - 9.02e3T + 2.07e9T^{2}$$
79 $$1 - 3.14e3T + 3.07e9T^{2}$$
83 $$1 + 4.27e4T + 3.93e9T^{2}$$
89 $$1 - 8.32e4T + 5.58e9T^{2}$$
97 $$1 + 4.25e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$