# Properties

 Label 2-1045-1.1-c3-0-177 Degree $2$ Conductor $1045$ Sign $-1$ Analytic cond. $61.6569$ Root an. cond. $7.85219$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.13·2-s + 1.57·3-s + 18.3·4-s − 5·5-s + 8.07·6-s − 16.5·7-s + 52.9·8-s − 24.5·9-s − 25.6·10-s − 11·11-s + 28.8·12-s − 24.1·13-s − 84.9·14-s − 7.87·15-s + 125.·16-s − 68.6·17-s − 125.·18-s + 19·19-s − 91.6·20-s − 26.0·21-s − 56.4·22-s − 112.·23-s + 83.3·24-s + 25·25-s − 123.·26-s − 81.1·27-s − 303.·28-s + ⋯
 L(s)  = 1 + 1.81·2-s + 0.303·3-s + 2.29·4-s − 0.447·5-s + 0.549·6-s − 0.893·7-s + 2.33·8-s − 0.908·9-s − 0.811·10-s − 0.301·11-s + 0.694·12-s − 0.515·13-s − 1.62·14-s − 0.135·15-s + 1.95·16-s − 0.979·17-s − 1.64·18-s + 0.229·19-s − 1.02·20-s − 0.270·21-s − 0.546·22-s − 1.02·23-s + 0.709·24-s + 0.200·25-s − 0.934·26-s − 0.578·27-s − 2.04·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $-1$ Analytic conductor: $$61.6569$$ Root analytic conductor: $$7.85219$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1045,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + 5T$$
11 $$1 + 11T$$
19 $$1 - 19T$$
good2 $$1 - 5.13T + 8T^{2}$$
3 $$1 - 1.57T + 27T^{2}$$
7 $$1 + 16.5T + 343T^{2}$$
13 $$1 + 24.1T + 2.19e3T^{2}$$
17 $$1 + 68.6T + 4.91e3T^{2}$$
23 $$1 + 112.T + 1.21e4T^{2}$$
29 $$1 - 97.6T + 2.43e4T^{2}$$
31 $$1 + 33.3T + 2.97e4T^{2}$$
37 $$1 + 133.T + 5.06e4T^{2}$$
41 $$1 + 244.T + 6.89e4T^{2}$$
43 $$1 - 383.T + 7.95e4T^{2}$$
47 $$1 + 70.4T + 1.03e5T^{2}$$
53 $$1 - 281.T + 1.48e5T^{2}$$
59 $$1 - 32.3T + 2.05e5T^{2}$$
61 $$1 + 411.T + 2.26e5T^{2}$$
67 $$1 + 202.T + 3.00e5T^{2}$$
71 $$1 - 616.T + 3.57e5T^{2}$$
73 $$1 - 12.4T + 3.89e5T^{2}$$
79 $$1 + 3.23T + 4.93e5T^{2}$$
83 $$1 + 304.T + 5.71e5T^{2}$$
89 $$1 + 706.T + 7.04e5T^{2}$$
97 $$1 - 1.80e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$