# Properties

 Label 2-1008-1.1-c3-0-30 Degree $2$ Conductor $1008$ Sign $-1$ Analytic cond. $59.4739$ Root an. cond. $7.71193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 6·5-s − 7·7-s + 30·11-s + 2·13-s − 66·17-s + 52·19-s + 114·23-s − 89·25-s − 72·29-s + 196·31-s + 42·35-s − 286·37-s + 378·41-s − 164·43-s − 228·47-s + 49·49-s + 348·53-s − 180·55-s − 348·59-s − 106·61-s − 12·65-s − 596·67-s + 630·71-s − 1.04e3·73-s − 210·77-s + 88·79-s − 1.44e3·83-s + ⋯
 L(s)  = 1 − 0.536·5-s − 0.377·7-s + 0.822·11-s + 0.0426·13-s − 0.941·17-s + 0.627·19-s + 1.03·23-s − 0.711·25-s − 0.461·29-s + 1.13·31-s + 0.202·35-s − 1.27·37-s + 1.43·41-s − 0.581·43-s − 0.707·47-s + 1/7·49-s + 0.901·53-s − 0.441·55-s − 0.767·59-s − 0.222·61-s − 0.0228·65-s − 1.08·67-s + 1.05·71-s − 1.67·73-s − 0.310·77-s + 0.125·79-s − 1.90·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $-1$ Analytic conductor: $$59.4739$$ Root analytic conductor: $$7.71193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{1008} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1008,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + p T$$
good5 $$1 + 6 T + p^{3} T^{2}$$
11 $$1 - 30 T + p^{3} T^{2}$$
13 $$1 - 2 T + p^{3} T^{2}$$
17 $$1 + 66 T + p^{3} T^{2}$$
19 $$1 - 52 T + p^{3} T^{2}$$
23 $$1 - 114 T + p^{3} T^{2}$$
29 $$1 + 72 T + p^{3} T^{2}$$
31 $$1 - 196 T + p^{3} T^{2}$$
37 $$1 + 286 T + p^{3} T^{2}$$
41 $$1 - 378 T + p^{3} T^{2}$$
43 $$1 + 164 T + p^{3} T^{2}$$
47 $$1 + 228 T + p^{3} T^{2}$$
53 $$1 - 348 T + p^{3} T^{2}$$
59 $$1 + 348 T + p^{3} T^{2}$$
61 $$1 + 106 T + p^{3} T^{2}$$
67 $$1 + 596 T + p^{3} T^{2}$$
71 $$1 - 630 T + p^{3} T^{2}$$
73 $$1 + 1042 T + p^{3} T^{2}$$
79 $$1 - 88 T + p^{3} T^{2}$$
83 $$1 + 1440 T + p^{3} T^{2}$$
89 $$1 + 1374 T + p^{3} T^{2}$$
97 $$1 + 34 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$