Properties

Degree $2$
Conductor $144$
Sign $-0.256 - 0.966i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.263 + 1.38i)2-s + (−1.86 − 0.731i)4-s + (2.63 + 2.63i)5-s − 0.207·7-s + (1.50 − 2.39i)8-s + (−4.35 + 2.96i)10-s + (−3.66 + 3.66i)11-s + (0.255 + 0.255i)13-s + (0.0545 − 0.287i)14-s + (2.93 + 2.72i)16-s − 0.654i·17-s + (4.46 − 4.46i)19-s + (−2.97 − 6.82i)20-s + (−4.13 − 6.06i)22-s − 3.48i·23-s + ⋯
L(s)  = 1  + (−0.186 + 0.982i)2-s + (−0.930 − 0.365i)4-s + (1.17 + 1.17i)5-s − 0.0783·7-s + (0.532 − 0.846i)8-s + (−1.37 + 0.937i)10-s + (−1.10 + 1.10i)11-s + (0.0708 + 0.0708i)13-s + (0.0145 − 0.0769i)14-s + (0.732 + 0.680i)16-s − 0.158i·17-s + (1.02 − 1.02i)19-s + (−0.665 − 1.52i)20-s + (−0.880 − 1.29i)22-s − 0.727i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.256 - 0.966i$
Motivic weight: \(1\)
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.256 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640331 + 0.832439i\)
\(L(\frac12)\) \(\approx\) \(0.640331 + 0.832439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.263 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (-2.63 - 2.63i)T + 5iT^{2} \)
7 \( 1 + 0.207T + 7T^{2} \)
11 \( 1 + (3.66 - 3.66i)T - 11iT^{2} \)
13 \( 1 + (-0.255 - 0.255i)T + 13iT^{2} \)
17 \( 1 + 0.654iT - 17T^{2} \)
19 \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (-4.33 + 4.33i)T - 29iT^{2} \)
31 \( 1 + 6.16iT - 31T^{2} \)
37 \( 1 + (-4.39 + 4.39i)T - 37iT^{2} \)
41 \( 1 - 0.0684T + 41T^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 + (-1.51 - 1.51i)T + 53iT^{2} \)
59 \( 1 + (-2.53 + 2.53i)T - 59iT^{2} \)
61 \( 1 + (5.46 + 5.46i)T + 61iT^{2} \)
67 \( 1 + (4.77 - 4.77i)T - 67iT^{2} \)
71 \( 1 + 5.94iT - 71T^{2} \)
73 \( 1 - 6.93iT - 73T^{2} \)
79 \( 1 - 4.72iT - 79T^{2} \)
83 \( 1 + (-4.32 - 4.32i)T + 83iT^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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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

−13.60259381281221554222531714825, −12.86191299350273643193519858628, −11.05004784635588473105974315976, −9.962054534022757188895824575415, −9.516169256796812984281141550945, −7.83862837643230134838169671827, −6.92506824209638619887362271600, −5.98229665205193848511937027180, −4.78400399338981699230298105072, −2.58912882997263270749218077864, 1.36942597756664329621759116926, 3.12570831033476695127540879071, 4.98302056126904891775899373369, 5.76019628690684055918755454189, 8.014925046907785587475894246240, 8.851328984788366672743805090567, 9.829976159141976462251214955890, 10.60719128894500726132266003222, 11.90547560685962892168518972926, 12.85404776781374822434017565542

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