Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} \cdot 11^{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  − 3-s + 2·5-s + 9-s + 6·13-s − 2·15-s + 2·17-s − 4·19-s − 25-s − 27-s + 2·29-s + 8·31-s + 6·37-s − 6·39-s + 10·41-s − 4·43-s + 2·45-s − 8·47-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 10·61-s + 12·65-s + 12·67-s + 2·73-s + 75-s + 16·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 1.28·61-s + 1.48·65-s + 1.46·67-s + 0.234·73-s + 0.115·75-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(284592\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{284592} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 284592,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.604521464$
$L(\frac12)$  $\approx$  $3.604521464$
$L(\frac{3}{2})$   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 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.74438712158307, −12.38422056918343, −11.65794296182368, −11.39258497700430, −10.86526068351536, −10.49852117292280, −9.974742852542660, −9.640706605996850, −9.091518352155373, −8.560906340943655, −8.044739042065425, −7.790664432485835, −6.775811553717064, −6.536542956233058, −6.152888456752462, −5.662294908651751, −5.328036965713202, −4.458309942244737, −4.242528685398361, −3.537732964596337, −2.923728142743918, −2.284848747401617, −1.692611829040588, −1.080755537225800, −0.5946490234043475, 0.5946490234043475, 1.080755537225800, 1.692611829040588, 2.284848747401617, 2.923728142743918, 3.537732964596337, 4.242528685398361, 4.458309942244737, 5.328036965713202, 5.662294908651751, 6.152888456752462, 6.536542956233058, 6.775811553717064, 7.790664432485835, 8.044739042065425, 8.560906340943655, 9.091518352155373, 9.640706605996850, 9.974742852542660, 10.49852117292280, 10.86526068351536, 11.39258497700430, 11.65794296182368, 12.38422056918343, 12.74438712158307

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