# Properties

 Label 2-1045-1.1-c3-0-176 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 + 3.55·2-s + 6.62·3-s + 4.64·4-s − 5·5-s + 23.5·6-s − 2.03·7-s − 11.9·8-s + 16.8·9-s − 17.7·10-s − 11·11-s + 30.7·12-s − 18.7·13-s − 7.22·14-s − 33.1·15-s − 79.5·16-s − 128.·17-s + 59.9·18-s + 19·19-s − 23.2·20-s − 13.4·21-s − 39.1·22-s + 68.3·23-s − 79.0·24-s + 25·25-s − 66.7·26-s − 67.0·27-s − 9.43·28-s + ⋯
 L(s)  = 1 + 1.25·2-s + 1.27·3-s + 0.580·4-s − 0.447·5-s + 1.60·6-s − 0.109·7-s − 0.527·8-s + 0.624·9-s − 0.562·10-s − 0.301·11-s + 0.740·12-s − 0.400·13-s − 0.137·14-s − 0.570·15-s − 1.24·16-s − 1.83·17-s + 0.785·18-s + 0.229·19-s − 0.259·20-s − 0.139·21-s − 0.379·22-s + 0.620·23-s − 0.672·24-s + 0.200·25-s − 0.503·26-s − 0.478·27-s − 0.0636·28-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 - 3.55T + 8T^{2}$$
3 $$1 - 6.62T + 27T^{2}$$
7 $$1 + 2.03T + 343T^{2}$$
13 $$1 + 18.7T + 2.19e3T^{2}$$
17 $$1 + 128.T + 4.91e3T^{2}$$
23 $$1 - 68.3T + 1.21e4T^{2}$$
29 $$1 + 89.8T + 2.43e4T^{2}$$
31 $$1 + 43.4T + 2.97e4T^{2}$$
37 $$1 + 142.T + 5.06e4T^{2}$$
41 $$1 - 367.T + 6.89e4T^{2}$$
43 $$1 + 298.T + 7.95e4T^{2}$$
47 $$1 + 51.7T + 1.03e5T^{2}$$
53 $$1 - 35.2T + 1.48e5T^{2}$$
59 $$1 - 374.T + 2.05e5T^{2}$$
61 $$1 - 470.T + 2.26e5T^{2}$$
67 $$1 - 682.T + 3.00e5T^{2}$$
71 $$1 + 636.T + 3.57e5T^{2}$$
73 $$1 + 400.T + 3.89e5T^{2}$$
79 $$1 + 830.T + 4.93e5T^{2}$$
83 $$1 - 1.25e3T + 5.71e5T^{2}$$
89 $$1 + 512.T + 7.04e5T^{2}$$
97 $$1 + 575.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$