Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 11-s − 4·17-s + 6·19-s − 25-s + 8·29-s + 8·31-s + 4·35-s − 10·37-s + 8·41-s − 2·43-s − 8·47-s − 3·49-s + 2·53-s + 2·55-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 − 8·85-s − 14·89-s + 12·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.970·17-s + 1.37·19-s − 1/5·25-s + 1.48·29-s + 1.43·31-s + 0.676·35-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 + 0.269·55-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 − 0.867·85-s − 1.48·89-s + 1.23·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(66924\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{66924} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 66924,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.894623665$
$L(\frac12)$  $\approx$  $3.894623665$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
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

−14.17616438580563, −13.76551500312578, −13.29918396533338, −12.80685247854024, −11.98347325909222, −11.71574112741032, −11.29378167963263, −10.52902057397101, −10.10014001051158, −9.719943700143101, −9.051740425407283, −8.584579859931684, −8.096614251116317, −7.482051267886554, −6.746466145850872, −6.464535840200978, −5.748241620786543, −5.149308550115347, −4.767992331143067, −4.112030285169341, −3.311707877401570, −2.645093196223023, −2.019580431435786, −1.382607936172275, −0.6890765745413557, 0.6890765745413557, 1.382607936172275, 2.019580431435786, 2.645093196223023, 3.311707877401570, 4.112030285169341, 4.767992331143067, 5.149308550115347, 5.748241620786543, 6.464535840200978, 6.746466145850872, 7.482051267886554, 8.096614251116317, 8.584579859931684, 9.051740425407283, 9.719943700143101, 10.10014001051158, 10.52902057397101, 11.29378167963263, 11.71574112741032, 11.98347325909222, 12.80685247854024, 13.29918396533338, 13.76551500312578, 14.17616438580563

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