# Properties

 Degree 2 Conductor 101 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·3-s − 2·4-s − 5-s − 2·7-s + 9-s − 2·11-s + 4·12-s + 13-s + 2·15-s + 4·16-s + 3·17-s − 5·19-s + 2·20-s + 4·21-s + 23-s − 4·25-s + 4·27-s + 4·28-s − 4·29-s − 9·31-s + 4·33-s + 2·35-s − 2·36-s − 2·37-s − 2·39-s + 8·41-s − 8·43-s + ⋯
 L(s)  = 1 − 1.15·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.15·12-s + 0.277·13-s + 0.516·15-s + 16-s + 0.727·17-s − 1.14·19-s + 0.447·20-s + 0.872·21-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.755·28-s − 0.742·29-s − 1.61·31-s + 0.696·33-s + 0.338·35-s − 1/3·36-s − 0.328·37-s − 0.320·39-s + 1.24·41-s − 1.21·43-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$101$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{101} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 101,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 101$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 101$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad101 $$1 + T$$
good2 $$1 + p T^{2}$$
3 $$1 + 2 T + p T^{2}$$
5 $$1 + T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 - T + p T^{2}$$
29 $$1 + 4 T + p T^{2}$$
31 $$1 + 9 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 8 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 - 7 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 + 14 T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 - 13 T + p T^{2}$$
73 $$1 - 8 T + p T^{2}$$
79 $$1 + 9 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 - 14 T + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}