# Properties

 Label 2-126-1.1-c5-0-11 Degree $2$ Conductor $126$ Sign $-1$ Analytic cond. $20.2083$ Root an. cond. $4.49537$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s + 16·4-s − 26·5-s − 49·7-s + 64·8-s − 104·10-s − 664·11-s + 318·13-s − 196·14-s + 256·16-s − 1.58e3·17-s + 236·19-s − 416·20-s − 2.65e3·22-s − 2.21e3·23-s − 2.44e3·25-s + 1.27e3·26-s − 784·28-s + 4.95e3·29-s − 7.12e3·31-s + 1.02e3·32-s − 6.32e3·34-s + 1.27e3·35-s + 4.35e3·37-s + 944·38-s − 1.66e3·40-s − 1.05e4·41-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s − 0.465·5-s − 0.377·7-s + 0.353·8-s − 0.328·10-s − 1.65·11-s + 0.521·13-s − 0.267·14-s + 1/4·16-s − 1.32·17-s + 0.149·19-s − 0.232·20-s − 1.16·22-s − 0.871·23-s − 0.783·25-s + 0.369·26-s − 0.188·28-s + 1.09·29-s − 1.33·31-s + 0.176·32-s − 0.938·34-s + 0.175·35-s + 0.523·37-s + 0.106·38-s − 0.164·40-s − 0.979·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $-1$ Analytic conductor: $$20.2083$$ Root analytic conductor: $$4.49537$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 126,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - p^{2} T$$
3 $$1$$
7 $$1 + p^{2} T$$
good5 $$1 + 26 T + p^{5} T^{2}$$
11 $$1 + 664 T + p^{5} T^{2}$$
13 $$1 - 318 T + p^{5} T^{2}$$
17 $$1 + 1582 T + p^{5} T^{2}$$
19 $$1 - 236 T + p^{5} T^{2}$$
23 $$1 + 2212 T + p^{5} T^{2}$$
29 $$1 - 4954 T + p^{5} T^{2}$$
31 $$1 + 7128 T + p^{5} T^{2}$$
37 $$1 - 4358 T + p^{5} T^{2}$$
41 $$1 + 10542 T + p^{5} T^{2}$$
43 $$1 + 8452 T + p^{5} T^{2}$$
47 $$1 + 5352 T + p^{5} T^{2}$$
53 $$1 - 33354 T + p^{5} T^{2}$$
59 $$1 - 15436 T + p^{5} T^{2}$$
61 $$1 + 36762 T + p^{5} T^{2}$$
67 $$1 - 40972 T + p^{5} T^{2}$$
71 $$1 - 9092 T + p^{5} T^{2}$$
73 $$1 + 73454 T + p^{5} T^{2}$$
79 $$1 - 89400 T + p^{5} T^{2}$$
83 $$1 - 6428 T + p^{5} T^{2}$$
89 $$1 - 122658 T + p^{5} T^{2}$$
97 $$1 - 21370 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$