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Properties

Label 2-12e2-48.11-c3-0-17
Degree $2$
Conductor $144$
Sign $0.318 + 0.947i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0720 + 2.82i)2-s + (−7.98 + 0.407i)4-s + (2.40 − 2.40i)5-s − 11.7·7-s + (−1.72 − 22.5i)8-s + (6.98 + 6.63i)10-s + (−34.7 − 34.7i)11-s + (−3.17 + 3.17i)13-s + (−0.844 − 33.1i)14-s + (63.6 − 6.50i)16-s − 98.0i·17-s + (−15.9 − 15.9i)19-s + (−18.2 + 20.2i)20-s + (95.7 − 100. i)22-s − 69.6i·23-s + ⋯
L(s)  = 1  + (0.0254 + 0.999i)2-s + (−0.998 + 0.0509i)4-s + (0.215 − 0.215i)5-s − 0.632·7-s + (−0.0763 − 0.997i)8-s + (0.220 + 0.209i)10-s + (−0.952 − 0.952i)11-s + (−0.0678 + 0.0678i)13-s + (−0.0161 − 0.632i)14-s + (0.994 − 0.101i)16-s − 1.39i·17-s + (−0.192 − 0.192i)19-s + (−0.204 + 0.226i)20-s + (0.927 − 0.976i)22-s − 0.631i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.318 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.509892 - 0.366381i\)
\(L(\frac12)\) \(\approx\) \(0.509892 - 0.366381i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0720 - 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (-2.40 + 2.40i)T - 125iT^{2} \)
7 \( 1 + 11.7T + 343T^{2} \)
11 \( 1 + (34.7 + 34.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (3.17 - 3.17i)T - 2.19e3iT^{2} \)
17 \( 1 + 98.0iT - 4.91e3T^{2} \)
19 \( 1 + (15.9 + 15.9i)T + 6.85e3iT^{2} \)
23 \( 1 + 69.6iT - 1.21e4T^{2} \)
29 \( 1 + (15.9 + 15.9i)T + 2.43e4iT^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + (37.0 + 37.0i)T + 5.06e4iT^{2} \)
41 \( 1 - 59.3T + 6.89e4T^{2} \)
43 \( 1 + (241. - 241. i)T - 7.95e4iT^{2} \)
47 \( 1 + 395.T + 1.03e5T^{2} \)
53 \( 1 + (-458. + 458. i)T - 1.48e5iT^{2} \)
59 \( 1 + (257. + 257. i)T + 2.05e5iT^{2} \)
61 \( 1 + (373. - 373. i)T - 2.26e5iT^{2} \)
67 \( 1 + (648. + 648. i)T + 3.00e5iT^{2} \)
71 \( 1 - 787. iT - 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 + 382. iT - 4.93e5T^{2} \)
83 \( 1 + (491. - 491. i)T - 5.71e5iT^{2} \)
89 \( 1 - 624.T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−12.95019996239600144365685202984, −11.44540205757545891309063633193, −10.05785305644596338895240713789, −9.170223089917369098519393759073, −8.123280468686131197112160661947, −6.99732247025537849288996313410, −5.86250943385193336071969074201, −4.84523782888891641166321967784, −3.14133012414340470793448501934, −0.29752365996545833082679648398, 1.93509618301740226994174820738, 3.33408151941353949688611745803, 4.74803235149854917018057117011, 6.12482144409897244629811191851, 7.72321309942857761170887335118, 8.929207115095834454144684511888, 10.19026617258628565316996821527, 10.46692015544797387074224011742, 11.96224723273424895046127105952, 12.75956416820896195440860149721

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