# Properties

 Degree 2 Conductor $2 \cdot 59$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 3·5-s + 6-s − 7-s − 8-s − 2·9-s + 3·10-s − 2·11-s − 12-s − 2·13-s + 14-s + 3·15-s + 16-s − 2·17-s + 2·18-s + 3·19-s − 3·20-s + 21-s + 2·22-s + 24-s + 4·25-s + 2·26-s + 5·27-s − 28-s − 29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s + 0.471·18-s + 0.688·19-s − 0.670·20-s + 0.218·21-s + 0.426·22-s + 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s − 0.188·28-s − 0.185·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$118$$    =    $$2 \cdot 59$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{118} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 118,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;59\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
59 $$1 + T$$
good3 $$1 + T + p T^{2}$$
5 $$1 + 3 T + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 - 3 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + T + p T^{2}$$
31 $$1 - 10 T + p T^{2}$$
37 $$1 + 12 T + p T^{2}$$
41 $$1 - 7 T + p T^{2}$$
43 $$1 + 6 T + p T^{2}$$
47 $$1 + 6 T + p T^{2}$$
53 $$1 + 11 T + p T^{2}$$
61 $$1 + 12 T + p T^{2}$$
67 $$1 - 10 T + p T^{2}$$
71 $$1 - 4 T + p T^{2}$$
73 $$1 - 12 T + p T^{2}$$
79 $$1 + 15 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 - 4 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}