# Properties

 Degree 2 Conductor $2^{2} \cdot 23$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s − 2·5-s − 4·7-s + 6·9-s + 2·11-s − 5·13-s + 6·15-s + 4·17-s − 2·19-s + 12·21-s + 23-s − 25-s − 9·27-s − 7·29-s − 3·31-s − 6·33-s + 8·35-s + 2·37-s + 15·39-s − 9·41-s − 8·43-s − 12·45-s + 9·47-s + 9·49-s − 12·51-s + 2·53-s − 4·55-s + ⋯
 L(s)  = 1 − 1.73·3-s − 0.894·5-s − 1.51·7-s + 2·9-s + 0.603·11-s − 1.38·13-s + 1.54·15-s + 0.970·17-s − 0.458·19-s + 2.61·21-s + 0.208·23-s − 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.538·31-s − 1.04·33-s + 1.35·35-s + 0.328·37-s + 2.40·39-s − 1.40·41-s − 1.21·43-s − 1.78·45-s + 1.31·47-s + 9/7·49-s − 1.68·51-s + 0.274·53-s − 0.539·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$92$$    =    $$2^{2} \cdot 23$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{92} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 92,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;23\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
23 $$1 - T$$
good3 $$1 + p T + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 + 5 T + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
29 $$1 + 7 T + p T^{2}$$
31 $$1 + 3 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 9 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 - 9 T + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 - 14 T + p T^{2}$$
71 $$1 + 3 T + p T^{2}$$
73 $$1 + 3 T + p T^{2}$$
79 $$1 + 6 T + p T^{2}$$
83 $$1 - 8 T + p T^{2}$$
89 $$1 - 12 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}