Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

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 more
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

Graph of the $Z$-function along the critical line