# Properties

 Degree 2 Conductor 18097 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s − 4·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 4·10-s − 6·11-s + 12-s − 5·13-s + 14-s + 4·15-s − 16-s − 7·17-s + 2·18-s − 4·19-s + 4·20-s + 21-s + 6·22-s − 4·23-s − 3·24-s + 11·25-s + 5·26-s + 5·27-s + 28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s − 1.80·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.917·19-s + 0.894·20-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.612·24-s + 11/5·25-s + 0.980·26-s + 0.962·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$18097$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{18097} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(2,\ 18097,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 18097$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 18097$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad18097 $$1 + T$$
good2 $$1 + T + p T^{2}$$
3 $$1 + T + p T^{2}$$
5 $$1 + 4 T + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 + 5 T + p T^{2}$$
17 $$1 + 7 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - T + p T^{2}$$
41 $$1 + 9 T + p T^{2}$$
43 $$1 - 2 T + p T^{2}$$
47 $$1 + 7 T + p T^{2}$$
53 $$1 + 9 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 + 15 T + p T^{2}$$
67 $$1 - 16 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 + 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}