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Properties

Label 2-12e2-16.3-c2-0-6
Degree $2$
Conductor $144$
Sign $0.412 - 0.911i$
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  + (−1.12 + 1.65i)2-s + (−1.47 − 3.71i)4-s + (0.0586 − 0.0586i)5-s + 4.61·7-s + (7.80 + 1.75i)8-s + (0.0310 + 0.162i)10-s + (5.36 + 5.36i)11-s + (11.0 + 11.0i)13-s + (−5.19 + 7.63i)14-s + (−11.6 + 10.9i)16-s + 12.8·17-s + (2.63 − 2.63i)19-s + (−0.304 − 0.131i)20-s + (−14.9 + 2.84i)22-s − 16.3·23-s + ⋯
L(s)  = 1  + (−0.562 + 0.826i)2-s + (−0.367 − 0.929i)4-s + (0.0117 − 0.0117i)5-s + 0.659·7-s + (0.975 + 0.218i)8-s + (0.00310 + 0.0162i)10-s + (0.487 + 0.487i)11-s + (0.850 + 0.850i)13-s + (−0.370 + 0.545i)14-s + (−0.729 + 0.683i)16-s + 0.757·17-s + (0.138 − 0.138i)19-s + (−0.0152 − 0.00659i)20-s + (−0.677 + 0.129i)22-s − 0.712·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.412 - 0.911i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.412 - 0.911i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.983481 + 0.634346i\)
\(L(\frac12)\) \(\approx\) \(0.983481 + 0.634346i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 1.65i)T \)
3 \( 1 \)
good5 \( 1 + (-0.0586 + 0.0586i)T - 25iT^{2} \)
7 \( 1 - 4.61T + 49T^{2} \)
11 \( 1 + (-5.36 - 5.36i)T + 121iT^{2} \)
13 \( 1 + (-11.0 - 11.0i)T + 169iT^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 + (-2.63 + 2.63i)T - 361iT^{2} \)
23 \( 1 + 16.3T + 529T^{2} \)
29 \( 1 + (-26.0 - 26.0i)T + 841iT^{2} \)
31 \( 1 + 20.2iT - 961T^{2} \)
37 \( 1 + (-41.2 + 41.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (0.786 + 0.786i)T + 1.84e3iT^{2} \)
47 \( 1 + 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (1.06 - 1.06i)T - 2.80e3iT^{2} \)
59 \( 1 + (32.5 + 32.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (-15.2 - 15.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (60.0 - 60.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 56.3T + 5.04e3T^{2} \)
73 \( 1 - 9.70iT - 5.32e3T^{2} \)
79 \( 1 + 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (26.7 - 26.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−13.34416434907926846933372119395, −11.86382552767939993248674512798, −10.89600937004736055148986527778, −9.715669304460743498238779610237, −8.807170565505234971496417946446, −7.76419760521022097874248386934, −6.71101019760727298429006340565, −5.50769028799827750079837184032, −4.18636931533633895672704997664, −1.51866660887037949973623794296, 1.16923206964364391420462402911, 3.04003759172283238292412603476, 4.43433460181694015089851240804, 6.11792080615838706809396421847, 7.88016116437806508800598492289, 8.432176768357095363614186011583, 9.758491570883789618527485968749, 10.66255217792787720282485650545, 11.58694940801149911579222155667, 12.39802207646820918708022594359

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