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Properties

Label 2-10944-1.1-c1-0-54
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 5·7-s + 5·11-s + 4·13-s + 3·17-s + 19-s + 4·25-s − 10·31-s + 15·35-s − 8·37-s − 5·43-s + 5·47-s + 18·49-s − 6·53-s + 15·55-s + 10·59-s + 5·61-s + 12·65-s − 10·67-s + 10·71-s − 11·73-s + 25·77-s + 10·79-s + 9·85-s + 10·89-s + 20·91-s + 3·95-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.88·7-s + 1.50·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s + 4/5·25-s − 1.79·31-s + 2.53·35-s − 1.31·37-s − 0.762·43-s + 0.729·47-s + 18/7·49-s − 0.824·53-s + 2.02·55-s + 1.30·59-s + 0.640·61-s + 1.48·65-s − 1.22·67-s + 1.18·71-s − 1.28·73-s + 2.84·77-s + 1.12·79-s + 0.976·85-s + 1.05·89-s + 2.09·91-s + 0.307·95-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: \(0\)
Selberg data: \((2,\ 10944,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.723898760\)
\(L(\frac12)\) \(\approx\) \(4.723898760\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.66722457858546, −16.09566551685759, −15.08966266947495, −14.63187617720987, −14.25775616532892, −13.77766009965843, −13.32482503605172, −12.36706184610846, −11.85977079150252, −11.22493211697853, −10.79665023509340, −10.12445383514757, −9.342063775119508, −8.877552297813950, −8.375775008840697, −7.575676337010137, −6.859713894237509, −6.149855353267183, −5.449076201032891, −5.130277105390104, −4.075429589183822, −3.540932996367122, −2.229999346272151, −1.496443743528067, −1.269509476352266, 1.269509476352266, 1.496443743528067, 2.229999346272151, 3.540932996367122, 4.075429589183822, 5.130277105390104, 5.449076201032891, 6.149855353267183, 6.859713894237509, 7.575676337010137, 8.375775008840697, 8.877552297813950, 9.342063775119508, 10.12445383514757, 10.79665023509340, 11.22493211697853, 11.85977079150252, 12.36706184610846, 13.32482503605172, 13.77766009965843, 14.25775616532892, 14.63187617720987, 15.08966266947495, 16.09566551685759, 16.66722457858546

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