Properties

Label 2-12e2-4.3-c8-0-17
Degree $2$
Conductor $144$
Sign $-1$
Analytic cond. $58.6625$
Root an. cond. $7.65914$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 726·5-s − 3.05e3i·7-s − 1.32e4i·11-s + 3.90e4·13-s + 6.58e4·17-s − 1.30e5i·19-s + 5.02e5i·23-s + 1.36e5·25-s − 2.02e5·29-s − 1.19e6i·31-s + 2.21e6i·35-s − 1.87e6·37-s − 3.09e6·41-s − 2.26e6i·43-s − 6.35e6i·47-s + ⋯
L(s)  = 1  − 1.16·5-s − 1.27i·7-s − 0.907i·11-s + 1.36·13-s + 0.787·17-s − 0.999i·19-s + 1.79i·23-s + 0.349·25-s − 0.285·29-s − 1.29i·31-s + 1.47i·35-s − 1.00·37-s − 1.09·41-s − 0.662i·43-s − 1.30i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(58.6625\)
Root analytic conductor: \(7.65914\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.7175438406\)
\(L(\frac12)\) \(\approx\) \(0.7175438406\)
\(L(5)\) 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 \( 1 + 726T + 3.90e5T^{2} \)
7 \( 1 + 3.05e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.32e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.90e4T + 8.15e8T^{2} \)
17 \( 1 - 6.58e4T + 6.97e9T^{2} \)
19 \( 1 + 1.30e5iT - 1.69e10T^{2} \)
23 \( 1 - 5.02e5iT - 7.83e10T^{2} \)
29 \( 1 + 2.02e5T + 5.00e11T^{2} \)
31 \( 1 + 1.19e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.87e6T + 3.51e12T^{2} \)
41 \( 1 + 3.09e6T + 7.98e12T^{2} \)
43 \( 1 + 2.26e6iT - 1.16e13T^{2} \)
47 \( 1 + 6.35e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.06e6T + 6.22e13T^{2} \)
59 \( 1 - 5.76e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.71e7T + 1.91e14T^{2} \)
67 \( 1 - 2.74e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.98e7iT - 6.45e14T^{2} \)
73 \( 1 + 5.32e7T + 8.06e14T^{2} \)
79 \( 1 + 1.82e7iT - 1.51e15T^{2} \)
83 \( 1 - 7.78e6iT - 2.25e15T^{2} \)
89 \( 1 + 8.66e7T + 3.93e15T^{2} \)
97 \( 1 + 7.39e7T + 7.83e15T^{2} \)
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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

−11.23588028393465378454250933998, −10.22760799966659563517862059299, −8.785302677587838440953240196590, −7.81770812513077481574615372421, −6.99894372701087408169432808351, −5.54566028193437022397373300271, −3.91848734507428364204258636077, −3.47396882675293710578096509076, −1.18703710312566985900498957512, −0.20706452938897119757122387540, 1.52895582506519606598232654234, 3.07669278746858495999326622779, 4.23190489262732324403069674012, 5.55663056819985722961458157232, 6.76856177848605040873031603738, 8.131624285403376019852162878292, 8.681876505504639995946102334657, 10.07991383198158764479110911142, 11.21474285187317208649225378273, 12.24332434150722792388088570233

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