Properties

Label 2-12e2-1.1-c3-0-1
Degree $2$
Conductor $144$
Sign $1$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 16·7-s + 12·11-s + 38·13-s + 126·17-s − 20·19-s + 168·23-s − 89·25-s − 30·29-s + 88·31-s − 96·35-s + 254·37-s − 42·41-s + 52·43-s − 96·47-s − 87·49-s − 198·53-s − 72·55-s − 660·59-s − 538·61-s − 228·65-s − 884·67-s + 792·71-s + 218·73-s + 192·77-s + 520·79-s − 492·83-s + ⋯
L(s)  = 1  − 0.536·5-s + 0.863·7-s + 0.328·11-s + 0.810·13-s + 1.79·17-s − 0.241·19-s + 1.52·23-s − 0.711·25-s − 0.192·29-s + 0.509·31-s − 0.463·35-s + 1.12·37-s − 0.159·41-s + 0.184·43-s − 0.297·47-s − 0.253·49-s − 0.513·53-s − 0.176·55-s − 1.45·59-s − 1.12·61-s − 0.435·65-s − 1.61·67-s + 1.32·71-s + 0.349·73-s + 0.284·77-s + 0.740·79-s − 0.650·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.765688700\)
\(L(\frac12)\) \(\approx\) \(1.765688700\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + 6 T + p^{3} T^{2} \)
7 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 - 126 T + p^{3} T^{2} \)
19 \( 1 + 20 T + p^{3} T^{2} \)
23 \( 1 - 168 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 - 88 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 + 42 T + p^{3} T^{2} \)
43 \( 1 - 52 T + p^{3} T^{2} \)
47 \( 1 + 96 T + p^{3} T^{2} \)
53 \( 1 + 198 T + p^{3} T^{2} \)
59 \( 1 + 660 T + p^{3} T^{2} \)
61 \( 1 + 538 T + p^{3} T^{2} \)
67 \( 1 + 884 T + p^{3} T^{2} \)
71 \( 1 - 792 T + p^{3} T^{2} \)
73 \( 1 - 218 T + p^{3} T^{2} \)
79 \( 1 - 520 T + p^{3} T^{2} \)
83 \( 1 + 492 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 - 1154 T + p^{3} T^{2} \)
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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

−12.49738205133274695699891667400, −11.56626804945871546126872322736, −10.80663310882223328799400979498, −9.490256807944533962424662346108, −8.278205257418518141117487675665, −7.50964792285464865739535101449, −6.01108322340042154271779382779, −4.69303630460735133463101893186, −3.34151852953240502395506080235, −1.24554233685785174047858765452, 1.24554233685785174047858765452, 3.34151852953240502395506080235, 4.69303630460735133463101893186, 6.01108322340042154271779382779, 7.50964792285464865739535101449, 8.278205257418518141117487675665, 9.490256807944533962424662346108, 10.80663310882223328799400979498, 11.56626804945871546126872322736, 12.49738205133274695699891667400

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