Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{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 + 2·7-s + 9-s − 2·13-s + 4·17-s − 6·19-s + 2·21-s + 27-s + 8·29-s + 8·31-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s − 8·47-s − 3·49-s + 4·51-s + 2·53-s − 6·57-s − 12·59-s − 10·61-s + 2·63-s + 12·67-s − 8·71-s + 6·73-s − 2·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 0.436·21-s + 0.192·27-s + 1.48·29-s + 1.43·31-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.794·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 1.46·67-s − 0.949·71-s + 0.702·73-s − 0.225·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(145200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{145200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 145200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.865434804$
$L(\frac12)$  $\approx$  $2.865434804$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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

−13.56095712602808, −12.76379829245913, −12.45780484652191, −12.01182522117037, −11.55263991912009, −10.91021852938024, −10.40593566444906, −9.980880217687691, −9.699845007436127, −8.729320140446758, −8.526135632443944, −8.176817078660355, −7.573250004336198, −7.079506203354342, −6.448777950637699, −6.099930439323659, −5.174004154010714, −4.815275572377314, −4.423080087756692, −3.682827112731010, −3.035883670965366, −2.639853932043619, −1.748579822853504, −1.481492211887340, −0.4668186260728744, 0.4668186260728744, 1.481492211887340, 1.748579822853504, 2.639853932043619, 3.035883670965366, 3.682827112731010, 4.423080087756692, 4.815275572377314, 5.174004154010714, 6.099930439323659, 6.448777950637699, 7.079506203354342, 7.573250004336198, 8.176817078660355, 8.526135632443944, 8.729320140446758, 9.699845007436127, 9.980880217687691, 10.40593566444906, 10.91021852938024, 11.55263991912009, 12.01182522117037, 12.45780484652191, 12.76379829245913, 13.56095712602808

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