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Properties

Label 2-12e2-16.11-c2-0-1
Degree $2$
Conductor $144$
Sign $-0.717 + 0.696i$
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  + (0.573 + 1.91i)2-s + (−3.34 + 2.19i)4-s + (−3.68 − 3.68i)5-s − 9.66·7-s + (−6.12 − 5.14i)8-s + (4.94 − 9.17i)10-s + (−5.51 + 5.51i)11-s + (−6.27 + 6.27i)13-s + (−5.53 − 18.5i)14-s + (6.35 − 14.6i)16-s + 6.78·17-s + (13.5 + 13.5i)19-s + (20.4 + 4.22i)20-s + (−13.7 − 7.41i)22-s − 17.0·23-s + ⋯
L(s)  = 1  + (0.286 + 0.958i)2-s + (−0.835 + 0.549i)4-s + (−0.737 − 0.737i)5-s − 1.38·7-s + (−0.765 − 0.643i)8-s + (0.494 − 0.917i)10-s + (−0.501 + 0.501i)11-s + (−0.482 + 0.482i)13-s + (−0.395 − 1.32i)14-s + (0.396 − 0.917i)16-s + 0.399·17-s + (0.711 + 0.711i)19-s + (1.02 + 0.211i)20-s + (−0.624 − 0.336i)22-s − 0.742·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.717 + 0.696i$
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: \((2,\ 144,\ (\ :1),\ -0.717 + 0.696i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0633998 - 0.156463i\)
\(L(\frac12)\) \(\approx\) \(0.0633998 - 0.156463i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.573 - 1.91i)T \)
3 \( 1 \)
good5 \( 1 + (3.68 + 3.68i)T + 25iT^{2} \)
7 \( 1 + 9.66T + 49T^{2} \)
11 \( 1 + (5.51 - 5.51i)T - 121iT^{2} \)
13 \( 1 + (6.27 - 6.27i)T - 169iT^{2} \)
17 \( 1 - 6.78T + 289T^{2} \)
19 \( 1 + (-13.5 - 13.5i)T + 361iT^{2} \)
23 \( 1 + 17.0T + 529T^{2} \)
29 \( 1 + (4.85 - 4.85i)T - 841iT^{2} \)
31 \( 1 - 5.25iT - 961T^{2} \)
37 \( 1 + (18.1 + 18.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (54.5 - 54.5i)T - 1.84e3iT^{2} \)
47 \( 1 - 40.4iT - 2.20e3T^{2} \)
53 \( 1 + (10.8 + 10.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (50.8 - 50.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (17.0 - 17.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-22.9 - 22.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 51.6T + 5.04e3T^{2} \)
73 \( 1 + 78.5iT - 5.32e3T^{2} \)
79 \( 1 + 108. iT - 6.24e3T^{2} \)
83 \( 1 + (57.3 + 57.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 44.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−13.47531784241180277553638183579, −12.48497719560158182187091133394, −12.08886386481646895125708677763, −10.05276132708642139609204923444, −9.240255072129433577957739768445, −8.034881185422090225906996289899, −7.13618538638625625724342545542, −5.90397729822676768941396341174, −4.62682744009068492294023668480, −3.41495026903471020384876688347, 0.096850873842387502485454755769, 2.87431317607089940919457236983, 3.61006054594482242283538638929, 5.35018006929628166230397902045, 6.71323850524147887331247068379, 8.081488822272467119328376179917, 9.541025401080719838887955212725, 10.26449082225609254846588265095, 11.28736863446669118563684869073, 12.17823689174794421378305528032

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