Properties

Degree $2$
Conductor $9792$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·11-s + 2·13-s − 17-s + 4·19-s − 25-s − 10·29-s − 8·31-s + 2·37-s − 10·41-s + 12·43-s − 7·49-s + 6·53-s − 8·55-s − 12·59-s + 10·61-s − 4·65-s − 12·67-s + 10·73-s + 8·79-s − 4·83-s + 2·85-s + 6·89-s − 8·95-s − 14·97-s − 10·101-s + 8·103-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.328·37-s − 1.56·41-s + 1.82·43-s − 49-s + 0.824·53-s − 1.07·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s + 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.820·95-s − 1.42·97-s − 0.995·101-s + 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9792\)    =    \(2^{6} \cdot 3^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{9792} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9792,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.41858973180027275469518461292, −6.75731369592298410865796616700, −5.95829119297760022471245554659, −5.33789404096089741989225038155, −4.36134938649640087809102403725, −3.72436468921055482710532071668, −3.36770001720799780980729067088, −2.04907435637041739111232790805, −1.20581906297715352620829947778, 0, 1.20581906297715352620829947778, 2.04907435637041739111232790805, 3.36770001720799780980729067088, 3.72436468921055482710532071668, 4.36134938649640087809102403725, 5.33789404096089741989225038155, 5.95829119297760022471245554659, 6.75731369592298410865796616700, 7.41858973180027275469518461292

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