Properties

 Degree 2 Conductor $3 \cdot 5$ Sign $1$ Motivic weight 2 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

 $$d$$ = $$2$$ $$N$$ = $$15$$    =    $$3 \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{15} (14, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 15,\ (\ :1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.747644$ $L(\frac12)$ $\approx$ $0.747644$ $L(2)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$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 )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}