# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 29$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s + 3·5-s + 6-s − 3·7-s − 8-s + 9-s − 3·10-s + 6·11-s − 12-s + 3·14-s − 3·15-s + 16-s + 7·17-s − 18-s + 5·19-s + 3·20-s + 3·21-s − 6·22-s − 8·23-s + 24-s + 4·25-s − 27-s − 3·28-s + 29-s + 3·30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s + 0.670·20-s + 0.654·21-s − 1.27·22-s − 1.66·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s − 0.566·28-s + 0.185·29-s + 0.547·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$174$$    =    $$2 \cdot 3 \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{174} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 174,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.8687758076$ $L(\frac12)$ $\approx$ $0.8687758076$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
3 $$1 + T$$
29 $$1 - T$$
good5 $$1 - 3 T + p T^{2}$$
7 $$1 + 3 T + p T^{2}$$
11 $$1 - 6 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 - 7 T + p T^{2}$$
19 $$1 - 5 T + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 3 T + p T^{2}$$
41 $$1 + 5 T + p T^{2}$$
43 $$1 - 3 T + p T^{2}$$
47 $$1 - 9 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 + 11 T + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 + 2 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 - 10 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}