Properties

Degree $2$
Conductor $144$
Sign $-0.982 - 0.187i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 2.71i)3-s + (−0.454 + 0.787i)5-s + (−6.10 + 3.52i)7-s + (−5.68 − 6.97i)9-s + (−6.96 + 4.02i)11-s + (3.35 − 5.81i)13-s + (−1.54 − 2.24i)15-s − 26.3·17-s + 20.5i·19-s + (−1.69 − 21.0i)21-s + (−21.8 − 12.6i)23-s + (12.0 + 20.9i)25-s + (26.2 − 6.44i)27-s + (15.1 + 26.2i)29-s + (−0.120 − 0.0693i)31-s + ⋯
L(s)  = 1  + (−0.428 + 0.903i)3-s + (−0.0909 + 0.157i)5-s + (−0.872 + 0.503i)7-s + (−0.632 − 0.774i)9-s + (−0.633 + 0.365i)11-s + (0.258 − 0.447i)13-s + (−0.103 − 0.149i)15-s − 1.54·17-s + 1.08i·19-s + (−0.0809 − 1.00i)21-s + (−0.949 − 0.547i)23-s + (0.483 + 0.837i)25-s + (0.971 − 0.238i)27-s + (0.523 + 0.906i)29-s + (−0.00387 − 0.00223i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.982 - 0.187i$
Motivic weight: \(2\)
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.982 - 0.187i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0510901 + 0.541112i\)
\(L(\frac12)\) \(\approx\) \(0.0510901 + 0.541112i\)
\(L(2)\) 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 \( 1 + (1.28 - 2.71i)T \)
good5 \( 1 + (0.454 - 0.787i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (6.10 - 3.52i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (6.96 - 4.02i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.35 + 5.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 26.3T + 289T^{2} \)
19 \( 1 - 20.5iT - 361T^{2} \)
23 \( 1 + (21.8 + 12.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.1 - 26.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (0.120 + 0.0693i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 69.7T + 1.36e3T^{2} \)
41 \( 1 + (29.3 - 50.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-2.45 + 1.41i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-70.7 + 40.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.0T + 2.80e3T^{2} \)
59 \( 1 + (77.1 + 44.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.0 - 41.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-44.0 - 25.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 68.4iT - 5.04e3T^{2} \)
73 \( 1 + 22.1T + 5.32e3T^{2} \)
79 \( 1 + (34.4 - 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (23.0 - 13.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 25.7T + 7.92e3T^{2} \)
97 \( 1 + (52.3 + 90.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.14021636318848711454038959659, −12.32354466365819744137344868613, −11.14757717663023988930576060390, −10.28763463041257430709898680913, −9.406710164896197942508802288078, −8.304453587773087047580399519759, −6.62976830844164196475889401253, −5.65020988964103856431851396673, −4.29874098430125133403917072557, −2.88519485308456242758633821767, 0.35312018984750964854082349129, 2.52316282509141612422920509808, 4.44102966219401460475918758168, 6.05801807463854426639284671297, 6.85026305021668154454389709438, 8.009158958995423455753904628005, 9.185523879111665042053223925082, 10.57987504231144303188438913296, 11.41342483491616082896338653528, 12.51462999218460986777175713317

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