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Properties

Label 2-12e2-48.29-c2-0-12
Degree $2$
Conductor $144$
Sign $-0.304 + 0.952i$
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  + (0.126 + 1.99i)2-s + (−3.96 + 0.504i)4-s + (−5.14 − 5.14i)5-s + 7.48i·7-s + (−1.50 − 7.85i)8-s + (9.61 − 10.9i)10-s + (−11.6 − 11.6i)11-s + (−14.0 − 14.0i)13-s + (−14.9 + 0.945i)14-s + (15.4 − 4.00i)16-s + 7.92i·17-s + (10.7 + 10.7i)19-s + (23.0 + 17.8i)20-s + (21.8 − 24.7i)22-s − 2.09·23-s + ⋯
L(s)  = 1  + (0.0631 + 0.998i)2-s + (−0.992 + 0.126i)4-s + (−1.02 − 1.02i)5-s + 1.06i·7-s + (−0.188 − 0.982i)8-s + (0.961 − 1.09i)10-s + (−1.06 − 1.06i)11-s + (−1.08 − 1.08i)13-s + (−1.06 + 0.0675i)14-s + (0.968 − 0.250i)16-s + 0.466i·17-s + (0.567 + 0.567i)19-s + (1.15 + 0.890i)20-s + (0.993 − 1.12i)22-s − 0.0909·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.304 + 0.952i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.304 + 0.952i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.110903 - 0.151953i\)
\(L(\frac12)\) \(\approx\) \(0.110903 - 0.151953i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.126 - 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (5.14 + 5.14i)T + 25iT^{2} \)
7 \( 1 - 7.48iT - 49T^{2} \)
11 \( 1 + (11.6 + 11.6i)T + 121iT^{2} \)
13 \( 1 + (14.0 + 14.0i)T + 169iT^{2} \)
17 \( 1 - 7.92iT - 289T^{2} \)
19 \( 1 + (-10.7 - 10.7i)T + 361iT^{2} \)
23 \( 1 + 2.09T + 529T^{2} \)
29 \( 1 + (23.6 - 23.6i)T - 841iT^{2} \)
31 \( 1 + 17.8T + 961T^{2} \)
37 \( 1 + (30.8 - 30.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 36.8T + 1.68e3T^{2} \)
43 \( 1 + (-28.4 + 28.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 65.3iT - 2.20e3T^{2} \)
53 \( 1 + (-9.05 - 9.05i)T + 2.80e3iT^{2} \)
59 \( 1 + (-74.0 - 74.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (53.4 + 53.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (20.6 + 20.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 39.6T + 5.04e3T^{2} \)
73 \( 1 + 91.3iT - 5.32e3T^{2} \)
79 \( 1 - 92.1T + 6.24e3T^{2} \)
83 \( 1 + (9.12 - 9.12i)T - 6.88e3iT^{2} \)
89 \( 1 + 63.2T + 7.92e3T^{2} \)
97 \( 1 - 152.T + 9.40e3T^{2} \)
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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

−12.57978766857532486050473257608, −12.00217541011537937637502234199, −10.35087689987344444548844770197, −8.926092163442396452357231774683, −8.270121377399418398946492276855, −7.49509816683234305082795311900, −5.60130719637656719512975587632, −5.11978050694882317947541797217, −3.41274805214356239196574108222, −0.11674781951570546180898443172, 2.45902156735126107045443414086, 3.86422288176278875246872756902, 4.86007181527798743313470776158, 7.17457749605534941534974033087, 7.63006069424214634275910051547, 9.451585748719611087373853303891, 10.31434042590512915502694529834, 11.17134677199340663500304972259, 11.93012885681470684207072089599, 13.01743237887369658787458542521

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