Properties

Label 2-12144-1.1-c1-0-18
Degree $2$
Conductor $12144$
Sign $1$
Analytic cond. $96.9703$
Root an. cond. $9.84735$
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-s + 4·7-s + 9-s − 11-s + 6·13-s − 2·17-s + 4·21-s + 23-s − 5·25-s + 27-s + 4·31-s − 33-s + 2·37-s + 6·39-s + 4·41-s − 4·43-s + 9·49-s − 2·51-s + 8·53-s − 4·59-s + 2·61-s + 4·63-s − 10·67-s + 69-s + 16·71-s − 10·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 0.872·21-s + 0.208·23-s − 25-s + 0.192·27-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.960·39-s + 0.624·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s + 1.09·53-s − 0.520·59-s + 0.256·61-s + 0.503·63-s − 1.22·67-s + 0.120·69-s + 1.89·71-s − 1.17·73-s − 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12144\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 23\)
Sign: $1$
Analytic conductor: \(96.9703\)
Root analytic conductor: \(9.84735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12144,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.784365378\)
\(L(\frac12)\) \(\approx\) \(3.784365378\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.22288053345283, −15.70183943072049, −15.13886989487546, −14.76735555420315, −13.98612776371657, −13.53829869921177, −13.29157530353897, −12.28436408552535, −11.73244134149859, −11.02754652663925, −10.82879487592251, −9.999546095945803, −9.202206440017978, −8.645802942417153, −8.099081378043382, −7.795411971421114, −6.902146278161043, −6.144616394217345, −5.477677293788279, −4.677950335738700, −4.106893821240556, −3.401916784328080, −2.392510008472681, −1.715790699024902, −0.9181399636161003, 0.9181399636161003, 1.715790699024902, 2.392510008472681, 3.401916784328080, 4.106893821240556, 4.677950335738700, 5.477677293788279, 6.144616394217345, 6.902146278161043, 7.795411971421114, 8.099081378043382, 8.645802942417153, 9.202206440017978, 9.999546095945803, 10.82879487592251, 11.02754652663925, 11.73244134149859, 12.28436408552535, 13.29157530353897, 13.53829869921177, 13.98612776371657, 14.76735555420315, 15.13886989487546, 15.70183943072049, 16.22288053345283

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