Properties

Label 2-12e2-9.7-c1-0-0
Degree $2$
Conductor $144$
Sign $0.173 - 0.984i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + (0.5 + 0.866i)5-s + (−1.5 + 2.59i)7-s − 2.99·9-s + (2.5 − 4.33i)11-s + (2.5 + 4.33i)13-s + (−1.49 + 0.866i)15-s − 2·17-s + 4·19-s + (−4.5 − 2.59i)21-s + (−0.5 − 0.866i)23-s + (2 − 3.46i)25-s − 5.19i·27-s + (4.5 − 7.79i)29-s + (−0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.223 + 0.387i)5-s + (−0.566 + 0.981i)7-s − 0.999·9-s + (0.753 − 1.30i)11-s + (0.693 + 1.20i)13-s + (−0.387 + 0.223i)15-s − 0.485·17-s + 0.917·19-s + (−0.981 − 0.566i)21-s + (−0.104 − 0.180i)23-s + (0.400 − 0.692i)25-s − 0.999i·27-s + (0.835 − 1.44i)29-s + (−0.0898 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834050 + 0.699851i\)
\(L(\frac12)\) \(\approx\) \(0.834050 + 0.699851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.69769288080972841376805078173, −11.94210221340455455198242576996, −11.37394362891619155388960776176, −10.20837879914991791089988814963, −9.110252047570485475361892622153, −8.608424029237451573602788670117, −6.49326918264950109464074853900, −5.76344677891560334785169615219, −4.10744132624351610669430925227, −2.83772857287000543069694881152, 1.31259399580731222414321464551, 3.39625860626815687212383917190, 5.17698809392735266441884198534, 6.64745874278211041824038420645, 7.33291629688830953280640226187, 8.610030166282449061877628348495, 9.780415696448317845299387661805, 10.89080209585435461437045072888, 12.17827624041516065345586035141, 12.91711883807172063042696979752

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