Properties

 Label 2-21e2-1.1-c5-0-15 Degree $2$ Conductor $441$ Sign $1$ Analytic cond. $70.7292$ Root an. cond. $8.41006$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 − 32·4-s + 427·13-s + 1.02e3·16-s − 3.14e3·19-s − 3.12e3·25-s − 2.72e3·31-s − 6.66e3·37-s + 2.24e4·43-s − 1.36e4·52-s + 3.86e4·61-s − 3.27e4·64-s − 3.79e4·67-s + 7.81e4·73-s + 1.00e5·76-s + 9.08e4·79-s + 1.34e5·97-s + 1.00e5·100-s + 2.11e5·103-s − 2.47e5·109-s + ⋯
 L(s)  = 1 − 4-s + 0.700·13-s + 16-s − 1.99·19-s − 25-s − 0.508·31-s − 0.799·37-s + 1.85·43-s − 0.700·52-s + 1.32·61-s − 64-s − 1.03·67-s + 1.71·73-s + 1.99·76-s + 1.63·79-s + 1.45·97-s + 100-s + 1.96·103-s − 1.99·109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$70.7292$$ Root analytic conductor: $$8.41006$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :5/2),\ 1)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.203144122$$ $$L(\frac12)$$ $$\approx$$ $$1.203144122$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + p^{5} T^{2}$$
5 $$1 + p^{5} T^{2}$$
11 $$1 + p^{5} T^{2}$$
13 $$1 - 427 T + p^{5} T^{2}$$
17 $$1 + p^{5} T^{2}$$
19 $$1 + 3143 T + p^{5} T^{2}$$
23 $$1 + p^{5} T^{2}$$
29 $$1 + p^{5} T^{2}$$
31 $$1 + 2723 T + p^{5} T^{2}$$
37 $$1 + 6661 T + p^{5} T^{2}$$
41 $$1 + p^{5} T^{2}$$
43 $$1 - 22475 T + p^{5} T^{2}$$
47 $$1 + p^{5} T^{2}$$
53 $$1 + p^{5} T^{2}$$
59 $$1 + p^{5} T^{2}$$
61 $$1 - 38626 T + p^{5} T^{2}$$
67 $$1 + 37939 T + p^{5} T^{2}$$
71 $$1 + p^{5} T^{2}$$
73 $$1 - 78127 T + p^{5} T^{2}$$
79 $$1 - 90857 T + p^{5} T^{2}$$
83 $$1 + p^{5} T^{2}$$
89 $$1 + p^{5} T^{2}$$
97 $$1 - 134386 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$