# Properties

 Label 2-15680-1.1-c1-0-50 Degree $2$ Conductor $15680$ Sign $1$ Analytic cond. $125.205$ Root an. cond. $11.1895$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s − 5-s − 2·9-s + 5·11-s + 7·13-s − 15-s + 3·17-s − 2·19-s + 8·23-s + 25-s − 5·27-s + 5·29-s + 10·31-s + 5·33-s − 4·37-s + 7·39-s + 6·41-s − 2·43-s + 2·45-s + 7·47-s + 3·51-s + 10·53-s − 5·55-s − 2·57-s − 10·59-s − 12·61-s − 7·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.50·11-s + 1.94·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 1.79·31-s + 0.870·33-s − 0.657·37-s + 1.12·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s + 1.02·47-s + 0.420·51-s + 1.37·53-s − 0.674·55-s − 0.264·57-s − 1.30·59-s − 1.53·61-s − 0.868·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$15680$$    =    $$2^{6} \cdot 5 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$125.205$$ Root analytic conductor: $$11.1895$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{15680} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 15680,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.459594375$$ $$L(\frac12)$$ $$\approx$$ $$3.459594375$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + T$$
7 $$1$$
good3 $$1 - T + p T^{2}$$
11 $$1 - 5 T + p T^{2}$$
13 $$1 - 7 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 - 5 T + p T^{2}$$
31 $$1 - 10 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 - 7 T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 + 12 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 7 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 8 T + p T^{2}$$
97 $$1 + 17 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$