Properties

Degree $2$
Conductor $144$
Sign $0.243 - 0.969i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.87 − 0.697i)2-s + (3.02 + 2.61i)4-s + (5.24 + 5.24i)5-s − 5.32·7-s + (−3.85 − 7.01i)8-s + (−6.17 − 13.4i)10-s + (−12.2 + 12.2i)11-s + (−5.73 + 5.73i)13-s + (9.98 + 3.71i)14-s + (2.33 + 15.8i)16-s + 23.3·17-s + (11.7 + 11.7i)19-s + (2.17 + 29.5i)20-s + (31.5 − 14.4i)22-s − 5.80·23-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)2-s + (0.757 + 0.653i)4-s + (1.04 + 1.04i)5-s − 0.761·7-s + (−0.481 − 0.876i)8-s + (−0.617 − 1.34i)10-s + (−1.11 + 1.11i)11-s + (−0.441 + 0.441i)13-s + (0.713 + 0.265i)14-s + (0.146 + 0.989i)16-s + 1.37·17-s + (0.618 + 0.618i)19-s + (0.108 + 1.47i)20-s + (1.43 − 0.657i)22-s − 0.252·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.243 - 0.969i$
Motivic weight: \(2\)
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.694100 + 0.541393i\)
\(L(\frac12)\) \(\approx\) \(0.694100 + 0.541393i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 + 0.697i)T \)
3 \( 1 \)
good5 \( 1 + (-5.24 - 5.24i)T + 25iT^{2} \)
7 \( 1 + 5.32T + 49T^{2} \)
11 \( 1 + (12.2 - 12.2i)T - 121iT^{2} \)
13 \( 1 + (5.73 - 5.73i)T - 169iT^{2} \)
17 \( 1 - 23.3T + 289T^{2} \)
19 \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \)
23 \( 1 + 5.80T + 529T^{2} \)
29 \( 1 + (18.3 - 18.3i)T - 841iT^{2} \)
31 \( 1 - 16.9iT - 961T^{2} \)
37 \( 1 + (-15.3 - 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (-66.9 - 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-27.1 + 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-65.2 + 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (37.6 + 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 - 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (24.1 + 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−12.84174693868091380389140516443, −12.03128934413930111734748420054, −10.51932975942703913554360819833, −10.07532737191160124113194384133, −9.376280071058669097279653715284, −7.65970668680825106195182778336, −6.91412534827963263430886338535, −5.65365524298172145440208653984, −3.23829373841088735411106114940, −2.08722869571862524433982341014, 0.76297326103762787394005385553, 2.73000152537449838374614107167, 5.41237667505948924686997216195, 5.89336514902468575432167746147, 7.55481144706274935005954415816, 8.544075682570909977111968107076, 9.647936922588478836167344664267, 10.09367486673953932345483122815, 11.46468951604371630613360675614, 12.82139054159663593407593257750

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