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Properties

Label 2-12e2-16.11-c4-0-36
Degree $2$
Conductor $144$
Sign $-0.819 - 0.573i$
Analytic cond. $14.8852$
Root an. cond. $3.85814$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.97 − 2.67i)2-s + (1.73 − 15.9i)4-s + (2.84 + 2.84i)5-s − 76.7·7-s + (−37.2 − 52.0i)8-s + (16.0 + 0.876i)10-s + (−121. + 121. i)11-s + (27.1 − 27.1i)13-s + (−228. + 205. i)14-s + (−249. − 55.2i)16-s + 88.0·17-s + (−261. − 261. i)19-s + (50.2 − 40.3i)20-s + (−37.3 + 686. i)22-s − 93.4·23-s + ⋯
L(s)  = 1  + (0.744 − 0.667i)2-s + (0.108 − 0.994i)4-s + (0.113 + 0.113i)5-s − 1.56·7-s + (−0.582 − 0.812i)8-s + (0.160 + 0.00876i)10-s + (−1.00 + 1.00i)11-s + (0.160 − 0.160i)13-s + (−1.16 + 1.04i)14-s + (−0.976 − 0.215i)16-s + 0.304·17-s + (−0.723 − 0.723i)19-s + (0.125 − 0.100i)20-s + (−0.0772 + 1.41i)22-s − 0.176·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(14.8852\)
Root analytic conductor: \(3.85814\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :2),\ -0.819 - 0.573i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.145005 + 0.460270i\)
\(L(\frac12)\) \(\approx\) \(0.145005 + 0.460270i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.97 + 2.67i)T \)
3 \( 1 \)
good5 \( 1 + (-2.84 - 2.84i)T + 625iT^{2} \)
7 \( 1 + 76.7T + 2.40e3T^{2} \)
11 \( 1 + (121. - 121. i)T - 1.46e4iT^{2} \)
13 \( 1 + (-27.1 + 27.1i)T - 2.85e4iT^{2} \)
17 \( 1 - 88.0T + 8.35e4T^{2} \)
19 \( 1 + (261. + 261. i)T + 1.30e5iT^{2} \)
23 \( 1 + 93.4T + 2.79e5T^{2} \)
29 \( 1 + (272. - 272. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (1.04e3 + 1.04e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 915. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.11e3 + 1.11e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (734. + 734. i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.20e3 + 1.20e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-580. + 580. i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.48e3 + 1.48e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 5.57e3T + 2.54e7T^{2} \)
73 \( 1 + 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.55e3 - 2.55e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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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

−12.15818207622230604125471008666, −10.58498878183551913886042711297, −10.12012442187768343398443195878, −9.018429410326602439883325111636, −7.13086295684187932913852567654, −6.16560015959152423894303746982, −4.88850361491513857296205031500, −3.44954593440434304466378904757, −2.31560487697406706104674077743, −0.13712263965820740763824016922, 2.82227905672313522555408653171, 3.87902660999495751349244115950, 5.59859777713100856511214848741, 6.26683874012451269636621573973, 7.54093505206554800563375253844, 8.663411743176057724102866859506, 9.854745456147501762962729782564, 11.13073643993942495175486234591, 12.42804803281566492831575059824, 13.14950613001385045978985396561

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