# Properties

 Label 2-15-15.14-c2-0-0 Degree $2$ Conductor $15$ Sign $1$ Analytic cond. $0.408720$ Root an. cond. $0.639312$ Motivic weight $2$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 3·3-s − 3·4-s − 5·5-s − 3·6-s + 7·8-s + 9·9-s + 5·10-s − 9·12-s − 15·15-s + 5·16-s + 14·17-s − 9·18-s − 22·19-s + 15·20-s − 34·23-s + 21·24-s + 25·25-s + 27·27-s + 15·30-s + 2·31-s − 33·32-s − 14·34-s − 27·36-s + 22·38-s − 35·40-s − 45·45-s + ⋯
 L(s)  = 1 − 1/2·2-s + 3-s − 3/4·4-s − 5-s − 1/2·6-s + 7/8·8-s + 9-s + 1/2·10-s − 3/4·12-s − 15-s + 5/16·16-s + 0.823·17-s − 1/2·18-s − 1.15·19-s + 3/4·20-s − 1.47·23-s + 7/8·24-s + 25-s + 27-s + 1/2·30-s + 2/31·31-s − 1.03·32-s − 0.411·34-s − 3/4·36-s + 0.578·38-s − 7/8·40-s − 45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$15$$    =    $$3 \cdot 5$$ Sign: $1$ Analytic conductor: $$0.408720$$ Root analytic conductor: $$0.639312$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: $\chi_{15} (14, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 15,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6668540075$$ $$L(\frac12)$$ $$\approx$$ $$0.6668540075$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - p T$$
5 $$1 + p T$$
good2 $$1 + T + p^{2} T^{2}$$
7 $$( 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}$$