Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
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  + 2·5-s − 7-s − 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s − 8·55-s + 6·61-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s + 16·79-s + 8·83-s + 12·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + 0.878·83-s + 1.30·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.874848804$
$L(\frac12)$  $\approx$  $1.874848804$
$L(\frac{3}{2})$   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 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−18.04421148172244, −17.41302863674042, −16.99205519175411, −16.35016377646882, −15.59820932067177, −15.06231706715128, −14.33674432553648, −13.67759199016125, −13.23433270317819, −12.33508176719082, −12.20709290106806, −10.92133072427255, −10.33280201779399, −9.978915405086881, −9.283586839942942, −8.272203299109376, −7.902596514850715, −6.843839268503023, −6.206692764635584, −5.501407586892367, −4.832973496128885, −3.833486670180882, −2.664513780841999, −2.240551987856353, −0.7654094440415861, 0.7654094440415861, 2.240551987856353, 2.664513780841999, 3.833486670180882, 4.832973496128885, 5.501407586892367, 6.206692764635584, 6.843839268503023, 7.902596514850715, 8.272203299109376, 9.283586839942942, 9.978915405086881, 10.33280201779399, 10.92133072427255, 12.20709290106806, 12.33508176719082, 13.23433270317819, 13.67759199016125, 14.33674432553648, 15.06231706715128, 15.59820932067177, 16.35016377646882, 16.99205519175411, 17.41302863674042, 18.04421148172244

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