Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{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  + 2·7-s − 11-s + 2·13-s + 4·17-s − 6·19-s + 8·29-s − 8·31-s − 10·37-s − 8·41-s + 2·43-s − 8·47-s − 3·49-s − 2·53-s − 12·59-s + 10·61-s − 12·67-s − 8·71-s − 6·73-s − 2·77-s − 2·79-s + 16·83-s + 14·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 1.48·29-s − 1.43·31-s − 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 0.949·71-s − 0.702·73-s − 0.227·77-s − 0.225·79-s + 1.75·83-s + 1.48·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9900,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−17.02094440328656, −16.23540159708448, −15.96858885114143, −14.99410598877249, −14.79074815448291, −14.09227730094487, −13.54957219104260, −12.87255492512033, −12.29374254719642, −11.73994411518079, −11.06048184203639, −10.41958761153053, −10.15879227420224, −9.041755745515047, −8.623789524799764, −8.016849869985653, −7.426293066105163, −6.576836520245016, −6.028364508478102, −5.114167094629317, −4.727931194896655, −3.735687555718937, −3.117890506524242, −1.999437633805573, −1.374725061447105, 0, 1.374725061447105, 1.999437633805573, 3.117890506524242, 3.735687555718937, 4.727931194896655, 5.114167094629317, 6.028364508478102, 6.576836520245016, 7.426293066105163, 8.016849869985653, 8.623789524799764, 9.041755745515047, 10.15879227420224, 10.41958761153053, 11.06048184203639, 11.73994411518079, 12.29374254719642, 12.87255492512033, 13.54957219104260, 14.09227730094487, 14.79074815448291, 14.99410598877249, 15.96858885114143, 16.23540159708448, 17.02094440328656

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