# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 3·5-s + 4·7-s + 6·9-s − 11-s − 3·13-s − 9·15-s + 2·17-s + 4·19-s − 12·21-s − 6·23-s + 4·25-s − 9·27-s − 29-s + 9·31-s + 3·33-s + 12·35-s − 8·37-s + 9·39-s − 8·41-s − 5·43-s + 18·45-s − 7·47-s + 9·49-s − 6·51-s − 5·53-s − 3·55-s + ⋯
 L(s)  = 1 − 1.73·3-s + 1.34·5-s + 1.51·7-s + 2·9-s − 0.301·11-s − 0.832·13-s − 2.32·15-s + 0.485·17-s + 0.917·19-s − 2.61·21-s − 1.25·23-s + 4/5·25-s − 1.73·27-s − 0.185·29-s + 1.61·31-s + 0.522·33-s + 2.02·35-s − 1.31·37-s + 1.44·39-s − 1.24·41-s − 0.762·43-s + 2.68·45-s − 1.02·47-s + 9/7·49-s − 0.840·51-s − 0.686·53-s − 0.404·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$116$$    =    $$2^{2} \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{116} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 116,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.8589770376$ $L(\frac12)$ $\approx$ $0.8589770376$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
29 $$1 + T$$
good3 $$1 + p T + p T^{2}$$
5 $$1 - 3 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 + T + p T^{2}$$
13 $$1 + 3 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
31 $$1 - 9 T + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 8 T + p T^{2}$$
43 $$1 + 5 T + p T^{2}$$
47 $$1 + 7 T + p T^{2}$$
53 $$1 + 5 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + T + p T^{2}$$
83 $$1 - 6 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}