Properties

Degree $2$
Conductor $6864$
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 − 2·5-s + 9-s + 11-s + 13-s + 2·15-s − 6·17-s + 4·19-s + 8·23-s − 25-s − 27-s − 10·29-s − 33-s + 6·37-s − 39-s + 10·41-s − 4·43-s − 2·45-s − 8·47-s − 7·49-s + 6·51-s − 10·53-s − 2·55-s − 4·57-s + 12·59-s + 14·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.174·33-s + 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.269·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6864\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6864} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049022738\)
\(L(\frac12)\) \(\approx\) \(1.049022738\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−17.38335752989767, −16.50022924791857, −16.01736062161417, −15.64866410906929, −14.79062662841783, −14.57634974427321, −13.34717090749148, −13.08886336965384, −12.49537775797773, −11.40624082888509, −11.31781876523701, −11.03715970424672, −9.744686640339811, −9.434145498060229, −8.555371669739059, −7.922331499607123, −7.106301351258225, −6.762604593997035, −5.807355979841270, −5.120839812501957, −4.330467788472542, −3.746505927835664, −2.835955623103920, −1.653231786711949, −0.5483736276693433, 0.5483736276693433, 1.653231786711949, 2.835955623103920, 3.746505927835664, 4.330467788472542, 5.120839812501957, 5.807355979841270, 6.762604593997035, 7.106301351258225, 7.922331499607123, 8.555371669739059, 9.434145498060229, 9.744686640339811, 11.03715970424672, 11.31781876523701, 11.40624082888509, 12.49537775797773, 13.08886336965384, 13.34717090749148, 14.57634974427321, 14.79062662841783, 15.64866410906929, 16.01736062161417, 16.50022924791857, 17.38335752989767

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