# Properties

 Degree 2 Conductor 751 Sign $1$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯
 L(s)  = 1 − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$751$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{751} (750, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 751,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.5381923100$ $L(\frac12)$ $\approx$ $0.5381923100$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 751$, $$F_p$$ is a polynomial of degree 2. If $p = 751$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad751 $$1 - T$$
good2 $$1 + T + T^{2}$$
3 $$1 - 1.41iT - T^{2}$$
5 $$1 + T^{2}$$
7 $$1 + 1.41iT - T^{2}$$
11 $$1 + 1.41iT - T^{2}$$
13 $$1 - T + T^{2}$$
17 $$1 + 1.41iT - T^{2}$$
19 $$1 - T + T^{2}$$
23 $$1 + T + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - T + T^{2}$$
41 $$1 - 1.41iT - T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - T + T^{2}$$
53 $$1 + T + T^{2}$$
59 $$1 + T + T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 + 1.41iT - T^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 + 1.41iT - T^{2}$$
89 $$1 - T + T^{2}$$
97 $$1 - T + T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}