# Properties

 Degree 2 Conductor $3 \cdot 5$ Sign $1$ Motivic weight 9 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 43.8·2-s + 81·3-s + 1.41e3·4-s − 625·5-s + 3.55e3·6-s − 7.86e3·7-s + 3.95e4·8-s + 6.56e3·9-s − 2.74e4·10-s − 4.93e4·11-s + 1.14e5·12-s + 2.42e4·13-s − 3.44e5·14-s − 5.06e4·15-s + 1.01e6·16-s + 2.68e5·17-s + 2.87e5·18-s − 1.68e5·19-s − 8.83e5·20-s − 6.36e5·21-s − 2.16e6·22-s − 2.12e6·23-s + 3.20e6·24-s + 3.90e5·25-s + 1.06e6·26-s + 5.31e5·27-s − 1.11e7·28-s + ⋯
 L(s)  = 1 + 1.93·2-s + 0.577·3-s + 2.76·4-s − 0.447·5-s + 1.11·6-s − 1.23·7-s + 3.41·8-s + 0.333·9-s − 0.867·10-s − 1.01·11-s + 1.59·12-s + 0.235·13-s − 2.40·14-s − 0.258·15-s + 3.86·16-s + 0.778·17-s + 0.646·18-s − 0.296·19-s − 1.23·20-s − 0.714·21-s − 1.97·22-s − 1.58·23-s + 1.97·24-s + 0.200·25-s + 0.456·26-s + 0.192·27-s − 3.41·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$15$$    =    $$3 \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$9$$ character : $\chi_{15} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 15,\ (\ :9/2),\ 1)$$ $$L(5)$$ $$\approx$$ $$4.99210$$ $$L(\frac12)$$ $$\approx$$ $$4.99210$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 - 81T$$
5 $$1 + 625T$$
good2 $$1 - 43.8T + 512T^{2}$$
7 $$1 + 7.86e3T + 4.03e7T^{2}$$
11 $$1 + 4.93e4T + 2.35e9T^{2}$$
13 $$1 - 2.42e4T + 1.06e10T^{2}$$
17 $$1 - 2.68e5T + 1.18e11T^{2}$$
19 $$1 + 1.68e5T + 3.22e11T^{2}$$
23 $$1 + 2.12e6T + 1.80e12T^{2}$$
29 $$1 - 3.89e5T + 1.45e13T^{2}$$
31 $$1 - 9.05e4T + 2.64e13T^{2}$$
37 $$1 + 3.31e6T + 1.29e14T^{2}$$
41 $$1 - 2.32e7T + 3.27e14T^{2}$$
43 $$1 - 1.91e7T + 5.02e14T^{2}$$
47 $$1 - 6.28e7T + 1.11e15T^{2}$$
53 $$1 + 1.80e5T + 3.29e15T^{2}$$
59 $$1 - 3.84e7T + 8.66e15T^{2}$$
61 $$1 + 5.53e5T + 1.16e16T^{2}$$
67 $$1 + 2.39e8T + 2.72e16T^{2}$$
71 $$1 - 1.28e8T + 4.58e16T^{2}$$
73 $$1 + 2.39e8T + 5.88e16T^{2}$$
79 $$1 + 5.28e8T + 1.19e17T^{2}$$
83 $$1 - 2.12e8T + 1.86e17T^{2}$$
89 $$1 + 2.07e8T + 3.50e17T^{2}$$
97 $$1 - 1.70e9T + 7.60e17T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}