↑

Properties

Label 2-10944-1.1-c1-0-29
Degree $2$
Conductor $10944$
Sign $-1$
Analytic cond. $87.3882$
Root an. cond. $9.34816$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 6·11-s + 4·13-s − 19-s + 4·23-s − 25-s + 6·29-s − 10·31-s + 8·35-s + 8·37-s + 6·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s + 12·55-s + 10·61-s − 8·65-s − 8·67-s + 8·71-s − 10·73-s + 24·77-s − 2·79-s − 6·83-s + 6·89-s − 16·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 1.80·11-s + 1.10·13-s − 0.229·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.79·31-s + 1.35·35-s + 1.31·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 1.61·55-s + 1.28·61-s − 0.992·65-s − 0.977·67-s + 0.949·71-s − 1.17·73-s + 2.73·77-s − 0.225·79-s − 0.658·83-s + 0.635·89-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10944\)    =    \(2^{6} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(87.3882\)
Root analytic conductor: \(9.34816\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10944,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−16.36684651772117, −16.16921689027712, −15.83789490860569, −15.24553516298223, −14.71537019026411, −13.67492663148547, −13.28264672495520, −12.78567189275586, −12.42692903180731, −11.53463090641116, −10.87107127228429, −10.54374423902097, −9.820796393949243, −9.103135172593510, −8.547565487666826, −7.801463334059470, −7.327603905170436, −6.636098278209057, −5.819911443959170, −5.410028030662741, −4.262613341337567, −3.782287164320862, −2.944618884910412, −2.478339654212683, −0.8781303465037501, 0, 0.8781303465037501, 2.478339654212683, 2.944618884910412, 3.782287164320862, 4.262613341337567, 5.410028030662741, 5.819911443959170, 6.636098278209057, 7.327603905170436, 7.801463334059470, 8.547565487666826, 9.103135172593510, 9.820796393949243, 10.54374423902097, 10.87107127228429, 11.53463090641116, 12.42692903180731, 12.78567189275586, 13.28264672495520, 13.67492663148547, 14.71537019026411, 15.24553516298223, 15.83789490860569, 16.16921689027712, 16.36684651772117

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