# Properties

 Degree 2 Conductor $5 \cdot 13 \cdot 31$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-s + ⋯
 L(s)  = 1 − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2015$$    =    $$5 \cdot 13 \cdot 31$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{2015} (2014, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 2015,\ (\ :0),\ 1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.4481774745$$ $$L(\frac12)$$ $$\approx$$ $$0.4481774745$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{5,\;13,\;31\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{5,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 - T$$
13 $$1 - T$$
31 $$1 - T$$
good2 $$1 + 1.94T + T^{2}$$
3 $$1 + 1.49T + T^{2}$$
7 $$1 - 1.13T + T^{2}$$
11 $$1 + 0.709T + T^{2}$$
17 $$1 - 1.77T + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - 0.241T + T^{2}$$
29 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.94T + T^{2}$$
47 $$1 - 0.241T + T^{2}$$
53 $$1 + 0.709T + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + 0.709T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - 1.13T + T^{2}$$
97 $$1 + 1.49T + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}