Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 6·13-s − 2·17-s − 4·19-s − 8·23-s − 6·29-s − 6·37-s − 10·41-s − 4·43-s + 8·47-s + 9·49-s − 10·53-s − 6·61-s − 4·67-s + 14·73-s + 16·79-s − 12·83-s − 2·89-s + 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1.11·29-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 0.768·61-s − 0.488·67-s + 1.63·73-s + 1.80·79-s − 1.31·83-s − 0.211·89-s + 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−16.80720830630348, −16.14248432806516, −15.42219812878591, −15.21796772427852, −14.43914108285470, −13.78036817218773, −13.36341450682060, −12.50323003153608, −12.40717842380479, −11.79643387074176, −10.86369794110369, −10.33336296024453, −9.702834638471090, −9.478311136752673, −8.657359924165229, −7.964992205116513, −7.235541924337734, −6.706265005381632, −6.185314000892176, −5.443320471623193, −4.694243514997047, −3.912708099054632, −3.305259296509431, −2.429125928192070, −1.854887955101878, 0, 0, 1.854887955101878, 2.429125928192070, 3.305259296509431, 3.912708099054632, 4.694243514997047, 5.443320471623193, 6.185314000892176, 6.706265005381632, 7.235541924337734, 7.964992205116513, 8.657359924165229, 9.478311136752673, 9.702834638471090, 10.33336296024453, 10.86369794110369, 11.79643387074176, 12.40717842380479, 12.50323003153608, 13.36341450682060, 13.78036817218773, 14.43914108285470, 15.21796772427852, 15.42219812878591, 16.14248432806516, 16.80720830630348

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