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Properties

Label 2-12e2-16.11-c2-0-14
Degree $2$
Conductor $144$
Sign $0.695 + 0.718i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.55 − 1.26i)2-s + (0.813 − 3.91i)4-s + (4.62 + 4.62i)5-s + 3.04·7-s + (−3.68 − 7.10i)8-s + (13.0 + 1.33i)10-s + (9.15 − 9.15i)11-s + (−5.78 + 5.78i)13-s + (4.72 − 3.84i)14-s + (−14.6 − 6.37i)16-s − 17.6·17-s + (−1.15 − 1.15i)19-s + (21.8 − 14.3i)20-s + (2.64 − 25.7i)22-s + 3.45·23-s + ⋯
L(s)  = 1  + (0.775 − 0.631i)2-s + (0.203 − 0.979i)4-s + (0.925 + 0.925i)5-s + 0.435·7-s + (−0.460 − 0.887i)8-s + (1.30 + 0.133i)10-s + (0.831 − 0.831i)11-s + (−0.444 + 0.444i)13-s + (0.337 − 0.274i)14-s + (−0.917 − 0.398i)16-s − 1.03·17-s + (−0.0606 − 0.0606i)19-s + (1.09 − 0.717i)20-s + (0.120 − 1.17i)22-s + 0.150·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.24987 - 0.954365i\)
\(L(\frac12)\) \(\approx\) \(2.24987 - 0.954365i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.55 + 1.26i)T \)
3 \( 1 \)
good5 \( 1 + (-4.62 - 4.62i)T + 25iT^{2} \)
7 \( 1 - 3.04T + 49T^{2} \)
11 \( 1 + (-9.15 + 9.15i)T - 121iT^{2} \)
13 \( 1 + (5.78 - 5.78i)T - 169iT^{2} \)
17 \( 1 + 17.6T + 289T^{2} \)
19 \( 1 + (1.15 + 1.15i)T + 361iT^{2} \)
23 \( 1 - 3.45T + 529T^{2} \)
29 \( 1 + (12.1 - 12.1i)T - 841iT^{2} \)
31 \( 1 - 38.5iT - 961T^{2} \)
37 \( 1 + (0.0972 + 0.0972i)T + 1.36e3iT^{2} \)
41 \( 1 - 51.5iT - 1.68e3T^{2} \)
43 \( 1 + (1.70 - 1.70i)T - 1.84e3iT^{2} \)
47 \( 1 + 24.1iT - 2.20e3T^{2} \)
53 \( 1 + (27.0 + 27.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (19.5 - 19.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (-16.7 + 16.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (75.8 + 75.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 134.T + 5.04e3T^{2} \)
73 \( 1 - 112. iT - 5.32e3T^{2} \)
79 \( 1 + 135. iT - 6.24e3T^{2} \)
83 \( 1 + (74.9 + 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 31.4iT - 7.92e3T^{2} \)
97 \( 1 - 31.5T + 9.40e3T^{2} \)
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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

−12.84130785351484482500087045156, −11.54040786255237857847226173064, −10.95394675203843828205633065686, −9.920448151458410802226336605171, −8.886652765481360467706693249541, −6.86803044647321976267550908896, −6.13436494669926282911731377279, −4.76433458850358004087322060750, −3.21422849125596634704803606306, −1.83788685373977806222383602296, 2.09122136111130729778017329646, 4.23827692263010062386885872374, 5.17941745152808114205905029326, 6.28539900866339501356000722513, 7.52430361426415390823920987655, 8.773756763068966715046911498647, 9.642967038376689108901501796536, 11.26233464731244481701879926746, 12.36636323536136630294952896782, 13.05724602874650107991900482562

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