Properties

Degree $2$
Conductor $235200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 5·11-s + 5·13-s − 4·17-s + 7·19-s + 23-s − 27-s − 2·31-s + 5·33-s + 37-s − 5·39-s − 5·41-s − 12·43-s + 11·47-s + 4·51-s − 9·53-s − 7·57-s − 4·59-s + 4·61-s + 12·67-s − 69-s − 2·71-s + 10·73-s + 12·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s + 1.38·13-s − 0.970·17-s + 1.60·19-s + 0.208·23-s − 0.192·27-s − 0.359·31-s + 0.870·33-s + 0.164·37-s − 0.800·39-s − 0.780·41-s − 1.82·43-s + 1.60·47-s + 0.560·51-s − 1.23·53-s − 0.927·57-s − 0.520·59-s + 0.512·61-s + 1.46·67-s − 0.120·69-s − 0.237·71-s + 1.17·73-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−13.23643021811658, −12.55791725985622, −12.38385025571814, −11.55059942276202, −11.22659866117361, −10.96305838265959, −10.45804976578246, −9.885009981343711, −9.561651372566315, −8.889140237288703, −8.327714953423856, −8.082822670831015, −7.375832606869338, −6.982157007170443, −6.454806821429160, −5.899377797271974, −5.362916220966629, −5.106496451575386, −4.513514653100581, −3.792807368130450, −3.320674383024143, −2.769568950958084, −2.048443133113772, −1.413602364319005, −0.7306684961840229, 0, 0.7306684961840229, 1.413602364319005, 2.048443133113772, 2.769568950958084, 3.320674383024143, 3.792807368130450, 4.513514653100581, 5.106496451575386, 5.362916220966629, 5.899377797271974, 6.454806821429160, 6.982157007170443, 7.375832606869338, 8.082822670831015, 8.327714953423856, 8.889140237288703, 9.561651372566315, 9.885009981343711, 10.45804976578246, 10.96305838265959, 11.22659866117361, 11.55059942276202, 12.38385025571814, 12.55791725985622, 13.23643021811658

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