# Properties

 Degree 2 Conductor $7 \cdot 11 \cdot 43$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯
 L(s)  = 1 + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3311$$    =    $$7 \cdot 11 \cdot 43$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{3311} (3310, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 3311,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $5.269367784$ $L(\frac12)$ $\approx$ $5.269367784$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{7,\;11,\;43\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{7,\;11,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 $$1 + T$$
11 $$1 + T$$
43 $$1 - T$$
good2 $$1 - 1.87T + T^{2}$$
3 $$1 - 1.28T + T^{2}$$
5 $$1 - 0.684T + T^{2}$$
13 $$1 + 1.73T + T^{2}$$
17 $$1 + 1.96T + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - T + T^{2}$$
29 $$1 - 1.53T + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - 1.96T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - 0.347T + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + 1.87T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - 1.28T + T^{2}$$
89 $$1 + T^{2}$$
97 $$1 - T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}