# Properties

 Label 2-1045-1.1-c3-0-141 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 + 4.96·2-s − 9.59·3-s + 16.6·4-s − 5·5-s − 47.6·6-s − 3.47·7-s + 42.8·8-s + 65.0·9-s − 24.8·10-s − 11·11-s − 159.·12-s + 11.2·13-s − 17.2·14-s + 47.9·15-s + 79.4·16-s − 2.14·17-s + 322.·18-s + 19·19-s − 83.1·20-s + 33.3·21-s − 54.5·22-s + 117.·23-s − 410.·24-s + 25·25-s + 55.7·26-s − 365.·27-s − 57.7·28-s + ⋯
 L(s)  = 1 + 1.75·2-s − 1.84·3-s + 2.07·4-s − 0.447·5-s − 3.23·6-s − 0.187·7-s + 1.89·8-s + 2.40·9-s − 0.784·10-s − 0.301·11-s − 3.83·12-s + 0.239·13-s − 0.328·14-s + 0.825·15-s + 1.24·16-s − 0.0306·17-s + 4.22·18-s + 0.229·19-s − 0.929·20-s + 0.346·21-s − 0.529·22-s + 1.06·23-s − 3.49·24-s + 0.200·25-s + 0.420·26-s − 2.60·27-s − 0.389·28-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 - 4.96T + 8T^{2}$$
3 $$1 + 9.59T + 27T^{2}$$
7 $$1 + 3.47T + 343T^{2}$$
13 $$1 - 11.2T + 2.19e3T^{2}$$
17 $$1 + 2.14T + 4.91e3T^{2}$$
23 $$1 - 117.T + 1.21e4T^{2}$$
29 $$1 + 57.3T + 2.43e4T^{2}$$
31 $$1 + 262.T + 2.97e4T^{2}$$
37 $$1 - 311.T + 5.06e4T^{2}$$
41 $$1 - 195.T + 6.89e4T^{2}$$
43 $$1 + 431.T + 7.95e4T^{2}$$
47 $$1 + 276.T + 1.03e5T^{2}$$
53 $$1 + 39.9T + 1.48e5T^{2}$$
59 $$1 + 618.T + 2.05e5T^{2}$$
61 $$1 + 44.3T + 2.26e5T^{2}$$
67 $$1 + 770.T + 3.00e5T^{2}$$
71 $$1 - 843.T + 3.57e5T^{2}$$
73 $$1 + 475.T + 3.89e5T^{2}$$
79 $$1 + 605.T + 4.93e5T^{2}$$
83 $$1 - 118.T + 5.71e5T^{2}$$
89 $$1 + 641.T + 7.04e5T^{2}$$
97 $$1 - 550.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$