# Properties

 Degree 2 Conductor 67 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s − 2·3-s + 2·4-s + 2·5-s − 4·6-s − 2·7-s + 9-s + 4·10-s − 4·11-s − 4·12-s + 2·13-s − 4·14-s − 4·15-s − 4·16-s + 3·17-s + 2·18-s + 7·19-s + 4·20-s + 4·21-s − 8·22-s + 9·23-s − 25-s + 4·26-s + 4·27-s − 4·28-s − 5·29-s − 8·30-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s + 0.554·13-s − 1.06·14-s − 1.03·15-s − 16-s + 0.727·17-s + 0.471·18-s + 1.60·19-s + 0.894·20-s + 0.872·21-s − 1.70·22-s + 1.87·23-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.755·28-s − 0.928·29-s − 1.46·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$67$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{67} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 67,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.273770036$ $L(\frac12)$ $\approx$ $1.273770036$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 67$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 67$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad67 $$1 - T$$
good2 $$1 - p T + p T^{2}$$
3 $$1 + 2 T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 - 7 T + p T^{2}$$
23 $$1 - 9 T + p T^{2}$$
29 $$1 + 5 T + p T^{2}$$
31 $$1 + 10 T + p T^{2}$$
37 $$1 + T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 + T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 - 9 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 7 T + p T^{2}$$
97 $$1 + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}