# Properties

 Label 2-5077-1.1-c1-0-410 Degree $2$ Conductor $5077$ Sign $-1$ Analytic cond. $40.5400$ Root an. cond. $6.36710$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $3$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 4·7-s + 6·9-s + 8·10-s − 6·11-s − 6·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 4·17-s − 12·18-s − 7·19-s − 8·20-s + 12·21-s + 12·22-s − 6·23-s + 11·25-s + 8·26-s − 9·27-s − 8·28-s − 6·29-s − 24·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s + 2·9-s + 2.52·10-s − 1.80·11-s − 1.73·12-s − 1.10·13-s + 2.13·14-s + 3.09·15-s − 16-s − 0.970·17-s − 2.82·18-s − 1.60·19-s − 1.78·20-s + 2.61·21-s + 2.55·22-s − 1.25·23-s + 11/5·25-s + 1.56·26-s − 1.73·27-s − 1.51·28-s − 1.11·29-s − 4.38·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5077$$ Sign: $-1$ Analytic conductor: $$40.5400$$ Root analytic conductor: $$6.36710$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(2,\ 5077,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad5077 $$1+O(T)$$
good2 $$1 + p T + p T^{2}$$
3 $$1 + p T + p T^{2}$$
5 $$1 + 4 T + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 2 T + p T^{2}$$
37 $$1 + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + 9 T + p T^{2}$$
53 $$1 + 9 T + p T^{2}$$
59 $$1 + 11 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + 14 T + p T^{2}$$
79 $$1 - 9 T + p T^{2}$$
83 $$1 + 2 T + p T^{2}$$
89 $$1 - 11 T + p T^{2}$$
97 $$1 - 6 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$