Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 11 $
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  + 3-s + 9-s + 11-s − 4·13-s + 6·17-s − 4·19-s + 6·23-s − 5·25-s + 27-s − 6·29-s − 8·31-s + 33-s + 10·37-s − 4·39-s − 6·41-s − 8·43-s + 6·47-s + 6·51-s − 4·57-s + 8·61-s + 4·67-s + 6·69-s + 6·71-s − 2·73-s − 5·75-s + 14·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 0.840·51-s − 0.529·57-s + 1.02·61-s + 0.488·67-s + 0.722·69-s + 0.712·71-s − 0.234·73-s − 0.577·75-s + 1.57·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(103488\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{103488} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 103488,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.507210066$
$L(\frac12)$  $\approx$  $2.507210066$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−13.80178474185762, −13.12146540395034, −12.78147106983752, −12.42739274805430, −11.73566870927841, −11.32688210435597, −10.80307017440866, −10.05150738662099, −9.792743227854839, −9.344196382752240, −8.769409814270528, −8.254433537544419, −7.591468912877096, −7.383246222230420, −6.795067494094719, −6.099103055183169, −5.491011832507313, −5.046399583168376, −4.397255548015523, −3.629725918864200, −3.434888906374553, −2.516428948461398, −2.071120265796813, −1.360866126229138, −0.4731487391469865, 0.4731487391469865, 1.360866126229138, 2.071120265796813, 2.516428948461398, 3.434888906374553, 3.629725918864200, 4.397255548015523, 5.046399583168376, 5.491011832507313, 6.099103055183169, 6.795067494094719, 7.383246222230420, 7.591468912877096, 8.254433537544419, 8.769409814270528, 9.344196382752240, 9.792743227854839, 10.05150738662099, 10.80307017440866, 11.32688210435597, 11.73566870927841, 12.42739274805430, 12.78147106983752, 13.12146540395034, 13.80178474185762

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