Properties

Degree $2$
Conductor $1176$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s − 6·11-s − 3·13-s − 2·15-s + 4·17-s − 5·19-s − 4·23-s − 25-s − 27-s − 4·29-s + 7·31-s + 6·33-s − 9·37-s + 3·39-s − 2·41-s − 43-s + 2·45-s + 2·47-s − 4·51-s + 8·53-s − 12·55-s + 5·57-s + 10·61-s − 6·65-s − 15·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.80·11-s − 0.832·13-s − 0.516·15-s + 0.970·17-s − 1.14·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.25·31-s + 1.04·33-s − 1.47·37-s + 0.480·39-s − 0.312·41-s − 0.152·43-s + 0.298·45-s + 0.291·47-s − 0.560·51-s + 1.09·53-s − 1.61·55-s + 0.662·57-s + 1.28·61-s − 0.744·65-s − 1.83·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−9.681721320131094163215273255189, −8.460071520116083697903537739907, −7.70587249118750730592766123774, −6.81232371947482166887069150763, −5.69453282174017204887014619372, −5.39310938366525563858610112111, −4.31058491195255984662115576742, −2.81439508368098570027009824752, −1.88941033513530829976904950599, 0, 1.88941033513530829976904950599, 2.81439508368098570027009824752, 4.31058491195255984662115576742, 5.39310938366525563858610112111, 5.69453282174017204887014619372, 6.81232371947482166887069150763, 7.70587249118750730592766123774, 8.460071520116083697903537739907, 9.681721320131094163215273255189

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