# Properties

 Degree 2 Conductor $3^{2} \cdot 11$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s − 4·5-s − 2·7-s + 3·8-s + 4·10-s − 11-s − 2·13-s + 2·14-s − 16-s + 2·17-s − 6·19-s + 4·20-s + 22-s + 4·23-s + 11·25-s + 2·26-s + 2·28-s − 6·29-s + 4·31-s − 5·32-s − 2·34-s + 8·35-s − 6·37-s + 6·38-s − 12·40-s − 10·41-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.755·7-s + 1.06·8-s + 1.26·10-s − 0.301·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s + 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s + 1.35·35-s − 0.986·37-s + 0.973·38-s − 1.89·40-s − 1.56·41-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$99$$    =    $$3^{2} \cdot 11$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{99} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 99,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;11\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1$$
11 $$1 + T$$
good2 $$1 + T + p T^{2}$$
5 $$1 + 4 T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + 6 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 + 6 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
show less
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−19.59706583976151, −19.18907849202123, −18.63161197571695, −17.19947188271381, −16.49569053124166, −15.51734692029922, −14.66161171979737, −13.13385760545446, −12.34290146894480, −11.12010949851049, −10.08905583189962, −8.821913560068405, −7.961683082550188, −6.973868727891961, −4.787978005940173, −3.547036332090546, 0, 3.547036332090546, 4.787978005940173, 6.973868727891961, 7.961683082550188, 8.821913560068405, 10.08905583189962, 11.12010949851049, 12.34290146894480, 13.13385760545446, 14.66161171979737, 15.51734692029922, 16.49569053124166, 17.19947188271381, 18.63161197571695, 19.18907849202123, 19.59706583976151