# Properties

 Degree 2 Conductor $3 \cdot 5$ Sign $1$ Motivic weight 7 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 15.7·2-s + 27·3-s + 120.·4-s + 125·5-s + 425.·6-s − 34.4·7-s − 121.·8-s + 729·9-s + 1.96e3·10-s − 3.96e3·11-s + 3.24e3·12-s + 5.60e3·13-s − 542.·14-s + 3.37e3·15-s − 1.73e4·16-s − 1.99e4·17-s + 1.14e4·18-s − 4.99e4·19-s + 1.50e4·20-s − 929.·21-s − 6.24e4·22-s + 1.09e5·23-s − 3.27e3·24-s + 1.56e4·25-s + 8.83e4·26-s + 1.96e4·27-s − 4.14e3·28-s + ⋯
 L(s)  = 1 + 1.39·2-s + 0.577·3-s + 0.939·4-s + 0.447·5-s + 0.804·6-s − 0.0379·7-s − 0.0837·8-s + 0.333·9-s + 0.622·10-s − 0.897·11-s + 0.542·12-s + 0.707·13-s − 0.0528·14-s + 0.258·15-s − 1.05·16-s − 0.982·17-s + 0.464·18-s − 1.67·19-s + 0.420·20-s − 0.0219·21-s − 1.25·22-s + 1.88·23-s − 0.0483·24-s + 0.199·25-s + 0.985·26-s + 0.192·27-s − 0.0356·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$15$$    =    $$3 \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : $\chi_{15} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 15,\ (\ :7/2),\ 1)$ $L(4)$ $\approx$ $3.32257$ $L(\frac12)$ $\approx$ $3.32257$ $L(\frac{9}{2})$ 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 - 27T$$
5 $$1 - 125T$$
good2 $$1 - 15.7T + 128T^{2}$$
7 $$1 + 34.4T + 8.23e5T^{2}$$
11 $$1 + 3.96e3T + 1.94e7T^{2}$$
13 $$1 - 5.60e3T + 6.27e7T^{2}$$
17 $$1 + 1.99e4T + 4.10e8T^{2}$$
19 $$1 + 4.99e4T + 8.93e8T^{2}$$
23 $$1 - 1.09e5T + 3.40e9T^{2}$$
29 $$1 - 1.92e5T + 1.72e10T^{2}$$
31 $$1 - 1.25e5T + 2.75e10T^{2}$$
37 $$1 + 7.43e4T + 9.49e10T^{2}$$
41 $$1 - 5.77e5T + 1.94e11T^{2}$$
43 $$1 - 2.64e5T + 2.71e11T^{2}$$
47 $$1 + 3.06e5T + 5.06e11T^{2}$$
53 $$1 + 4.46e5T + 1.17e12T^{2}$$
59 $$1 - 1.97e6T + 2.48e12T^{2}$$
61 $$1 + 1.27e6T + 3.14e12T^{2}$$
67 $$1 + 4.12e6T + 6.06e12T^{2}$$
71 $$1 + 2.81e6T + 9.09e12T^{2}$$
73 $$1 - 4.01e6T + 1.10e13T^{2}$$
79 $$1 + 1.32e6T + 1.92e13T^{2}$$
83 $$1 + 1.91e6T + 2.71e13T^{2}$$
89 $$1 - 8.00e6T + 4.42e13T^{2}$$
97 $$1 + 3.89e6T + 8.07e13T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}