Properties

Degree $2$
Conductor $111600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s − 6·13-s + 2·17-s − 4·19-s + 8·23-s − 6·29-s + 31-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s − 10·53-s − 12·59-s − 2·61-s − 4·67-s − 2·73-s − 4·83-s + 14·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1.11·29-s + 0.179·31-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s − 1.37·53-s − 1.56·59-s − 0.256·61-s − 0.488·67-s − 0.234·73-s − 0.439·83-s + 1.48·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{111600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 111600,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
31 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−14.19949916839340, −13.50305627425079, −13.06597432424097, −12.76437208499301, −12.25337389323461, −11.76026035767977, −11.11269476570169, −10.71200445524117, −10.25642793427591, −9.694766074091709, −9.335534889084470, −8.710345603916990, −8.055860755394801, −7.698191808680919, −7.225506002242970, −6.674291782215578, −6.107180361845443, −5.320740869958003, −4.940782905197712, −4.699896956689135, −3.768914416525847, −3.009851292432664, −2.750984035364851, −1.969633493836992, −1.339242890695639, 0, 0, 1.339242890695639, 1.969633493836992, 2.750984035364851, 3.009851292432664, 3.768914416525847, 4.699896956689135, 4.940782905197712, 5.320740869958003, 6.107180361845443, 6.674291782215578, 7.225506002242970, 7.698191808680919, 8.055860755394801, 8.710345603916990, 9.335534889084470, 9.694766074091709, 10.25642793427591, 10.71200445524117, 11.11269476570169, 11.76026035767977, 12.25337389323461, 12.76437208499301, 13.06597432424097, 13.50305627425079, 14.19949916839340

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