# Properties

 Label 2-7e2-1.1-c5-0-3 Degree $2$ Conductor $49$ Sign $1$ Analytic cond. $7.85880$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 10·2-s + 14·3-s + 68·4-s + 56·5-s − 140·6-s − 360·8-s − 47·9-s − 560·10-s + 232·11-s + 952·12-s + 140·13-s + 784·15-s + 1.42e3·16-s + 1.72e3·17-s + 470·18-s + 98·19-s + 3.80e3·20-s − 2.32e3·22-s + 1.82e3·23-s − 5.04e3·24-s + 11·25-s − 1.40e3·26-s − 4.06e3·27-s + 3.41e3·29-s − 7.84e3·30-s + 7.64e3·31-s − 2.72e3·32-s + ⋯
 L(s)  = 1 − 1.76·2-s + 0.898·3-s + 17/8·4-s + 1.00·5-s − 1.58·6-s − 1.98·8-s − 0.193·9-s − 1.77·10-s + 0.578·11-s + 1.90·12-s + 0.229·13-s + 0.899·15-s + 1.39·16-s + 1.44·17-s + 0.341·18-s + 0.0622·19-s + 2.12·20-s − 1.02·22-s + 0.718·23-s − 1.78·24-s + 0.00351·25-s − 0.406·26-s − 1.07·27-s + 0.754·29-s − 1.59·30-s + 1.42·31-s − 0.469·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $1$ Analytic conductor: $$7.85880$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.196461872$$ $$L(\frac12)$$ $$\approx$$ $$1.196461872$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
good2 $$1 + 5 p T + p^{5} T^{2}$$
3 $$1 - 14 T + p^{5} T^{2}$$
5 $$1 - 56 T + p^{5} T^{2}$$
11 $$1 - 232 T + p^{5} T^{2}$$
13 $$1 - 140 T + p^{5} T^{2}$$
17 $$1 - 1722 T + p^{5} T^{2}$$
19 $$1 - 98 T + p^{5} T^{2}$$
23 $$1 - 1824 T + p^{5} T^{2}$$
29 $$1 - 3418 T + p^{5} T^{2}$$
31 $$1 - 7644 T + p^{5} T^{2}$$
37 $$1 + 10398 T + p^{5} T^{2}$$
41 $$1 - 17962 T + p^{5} T^{2}$$
43 $$1 - 10880 T + p^{5} T^{2}$$
47 $$1 + 9324 T + p^{5} T^{2}$$
53 $$1 - 2262 T + p^{5} T^{2}$$
59 $$1 - 2730 T + p^{5} T^{2}$$
61 $$1 + 25648 T + p^{5} T^{2}$$
67 $$1 + 48404 T + p^{5} T^{2}$$
71 $$1 + 58560 T + p^{5} T^{2}$$
73 $$1 + 68082 T + p^{5} T^{2}$$
79 $$1 - 31784 T + p^{5} T^{2}$$
83 $$1 - 20538 T + p^{5} T^{2}$$
89 $$1 - 50582 T + p^{5} T^{2}$$
97 $$1 - 58506 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$