# Properties

 Degree 2 Conductor $3^{2} \cdot 17$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s + 3·11-s − 5·13-s − 4·14-s − 4·16-s + 17-s − 19-s + 2·20-s + 6·22-s + 7·23-s − 4·25-s − 10·26-s − 4·28-s − 6·29-s + 4·31-s − 8·32-s + 2·34-s − 2·35-s + 10·37-s − 2·38-s − 9·41-s + 43-s + 6·44-s + ⋯
 L(s)  = 1 + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s + 0.904·11-s − 1.38·13-s − 1.06·14-s − 16-s + 0.242·17-s − 0.229·19-s + 0.447·20-s + 1.27·22-s + 1.45·23-s − 4/5·25-s − 1.96·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 1.41·32-s + 0.342·34-s − 0.338·35-s + 1.64·37-s − 0.324·38-s − 1.40·41-s + 0.152·43-s + 0.904·44-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$153$$    =    $$3^{2} \cdot 17$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{153} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 153,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $2.076797451$ $L(\frac12)$ $\approx$ $2.076797451$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;17\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1$$
17 $$1 - T$$
good2 $$1 - p T + p T^{2}$$
5 $$1 - T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
11 $$1 - 3 T + p T^{2}$$
13 $$1 + 5 T + p T^{2}$$
19 $$1 + T + p T^{2}$$
23 $$1 - 7 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 + 9 T + p T^{2}$$
43 $$1 - T + p T^{2}$$
47 $$1 - 12 T + p T^{2}$$
53 $$1 - 12 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + 6 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 + 2 T + p T^{2}$$
97 $$1 - 8 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}