Properties

 Label 2-168e2-1.1-c1-0-125 Degree $2$ Conductor $28224$ Sign $-1$ Analytic cond. $225.369$ Root an. cond. $15.0123$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

Origins

Dirichlet series

 L(s)  = 1 + 4·13-s + 4·17-s − 4·19-s − 4·23-s − 5·25-s + 2·29-s − 8·31-s + 6·37-s + 12·41-s − 4·43-s − 8·47-s + 6·53-s − 12·59-s + 4·61-s + 4·67-s + 12·71-s − 8·73-s − 16·79-s + 4·83-s + 4·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
 L(s)  = 1 + 1.10·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s − 25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 1.87·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s − 1.56·59-s + 0.512·61-s + 0.488·67-s + 1.42·71-s − 0.936·73-s − 1.80·79-s + 0.439·83-s + 0.423·89-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$28224$$    =    $$2^{6} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$225.369$$ Root analytic conductor: $$15.0123$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{28224} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 28224,\ (\ :1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$$
good5 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 - 12 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 + 8 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 4 T + p T^{2}$$
97 $$1 + 16 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$