Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7^{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  − 4·13-s + 6·17-s − 2·19-s − 5·25-s − 6·29-s − 4·31-s − 2·37-s + 6·41-s + 8·43-s + 12·47-s + 6·53-s − 6·59-s + 8·61-s − 4·67-s − 2·73-s − 8·79-s − 6·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.10·13-s + 1.45·17-s − 0.458·19-s − 25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 0.824·53-s − 0.781·59-s + 1.02·61-s − 0.488·67-s − 0.234·73-s − 0.900·79-s − 0.658·83-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28224} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 28224,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.636221909$
$L(\frac12)$  $\approx$  $1.636221909$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−15.03327177988431, −14.76896600617999, −14.15128809036029, −13.77257359836199, −12.93864448079597, −12.52360167761477, −12.11037301117958, −11.50258775136387, −10.90283142321011, −10.27015925346340, −9.844773669544538, −9.237791516872057, −8.781498045916177, −7.839788365844663, −7.523590297051902, −7.114066372677540, −6.133020046741697, −5.586362923008411, −5.240546102453576, −4.166079274790088, −3.907439565531361, −2.918381259158318, −2.322934417029532, −1.516305065593208, −0.4946297798540870, 0.4946297798540870, 1.516305065593208, 2.322934417029532, 2.918381259158318, 3.907439565531361, 4.166079274790088, 5.240546102453576, 5.586362923008411, 6.133020046741697, 7.114066372677540, 7.523590297051902, 7.839788365844663, 8.781498045916177, 9.237791516872057, 9.844773669544538, 10.27015925346340, 10.90283142321011, 11.50258775136387, 12.11037301117958, 12.52360167761477, 12.93864448079597, 13.77257359836199, 14.15128809036029, 14.76896600617999, 15.03327177988431

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