Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·5-s + 9-s + 2·11-s − 6·13-s + 4·15-s + 4·17-s + 4·19-s − 2·23-s + 11·25-s + 27-s + 2·29-s + 2·33-s − 2·37-s − 6·39-s − 4·43-s + 4·45-s + 12·47-s + 4·51-s + 6·53-s + 8·55-s + 4·57-s + 8·59-s + 6·61-s − 24·65-s − 8·67-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s − 0.417·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s − 0.328·37-s − 0.960·39-s − 0.609·43-s + 0.596·45-s + 1.75·47-s + 0.560·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.04·59-s + 0.768·61-s − 2.97·65-s − 0.977·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.158422323\)
\(L(\frac12)\) \(\approx\) \(4.158422323\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−7.48055140088893310165370888362, −7.16129745430315884399569933958, −6.29720922441833452882432131874, −5.57231195325264994616573262274, −5.14056436828631480262325484346, −4.24974945413085103790315507050, −3.18166849287238476303908403717, −2.53627159495471071974969043910, −1.86729050078829565643852869820, −0.995983380836290695648376877184, 0.995983380836290695648376877184, 1.86729050078829565643852869820, 2.53627159495471071974969043910, 3.18166849287238476303908403717, 4.24974945413085103790315507050, 5.14056436828631480262325484346, 5.57231195325264994616573262274, 6.29720922441833452882432131874, 7.16129745430315884399569933958, 7.48055140088893310165370888362

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