# Properties

 Label 2-960-15.14-c2-0-65 Degree $2$ Conductor $960$ Sign $1$ Analytic cond. $26.1581$ Root an. cond. $5.11449$ Motivic weight $2$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3·3-s + 5·5-s + 9·9-s + 15·15-s + 14·17-s − 22·19-s + 34·23-s + 25·25-s + 27·27-s − 2·31-s + 45·45-s − 14·47-s + 49·49-s + 42·51-s − 86·53-s − 66·57-s + 118·61-s + 102·69-s + 75·75-s − 98·79-s + 81·81-s − 154·83-s + 70·85-s − 6·93-s − 110·95-s − 106·107-s + 22·109-s + ⋯
 L(s)  = 1 + 3-s + 5-s + 9-s + 15-s + 0.823·17-s − 1.15·19-s + 1.47·23-s + 25-s + 27-s − 0.0645·31-s + 45-s − 0.297·47-s + 49-s + 0.823·51-s − 1.62·53-s − 1.15·57-s + 1.93·61-s + 1.47·69-s + 75-s − 1.24·79-s + 81-s − 1.85·83-s + 0.823·85-s − 0.0645·93-s − 1.15·95-s − 0.990·107-s + 0.201·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$960$$    =    $$2^{6} \cdot 3 \cdot 5$$ Sign: $1$ Analytic conductor: $$26.1581$$ Root analytic conductor: $$5.11449$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: $\chi_{960} (449, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 960,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.535966122$$ $$L(\frac12)$$ $$\approx$$ $$3.535966122$$ $$L(2)$$ 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 - p T$$
5 $$1 - p T$$
good7 $$( 1 - p T )( 1 + p T )$$
11 $$( 1 - p T )( 1 + p T )$$
13 $$( 1 - p T )( 1 + p T )$$
17 $$1 - 14 T + p^{2} T^{2}$$
19 $$1 + 22 T + p^{2} T^{2}$$
23 $$1 - 34 T + p^{2} T^{2}$$
29 $$( 1 - p T )( 1 + p T )$$
31 $$1 + 2 T + p^{2} T^{2}$$
37 $$( 1 - p T )( 1 + p T )$$
41 $$( 1 - p T )( 1 + p T )$$
43 $$( 1 - p T )( 1 + p T )$$
47 $$1 + 14 T + p^{2} T^{2}$$
53 $$1 + 86 T + p^{2} T^{2}$$
59 $$( 1 - p T )( 1 + p T )$$
61 $$1 - 118 T + p^{2} T^{2}$$
67 $$( 1 - p T )( 1 + p T )$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$( 1 - p T )( 1 + p T )$$
79 $$1 + 98 T + p^{2} T^{2}$$
83 $$1 + 154 T + p^{2} T^{2}$$
89 $$( 1 - p T )( 1 + p T )$$
97 $$( 1 - p T )( 1 + p T )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$