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Properties

Label 2-12e2-16.11-c4-0-20
Degree $2$
Conductor $144$
Sign $0.245 + 0.969i$
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  + (−3.89 − 0.921i)2-s + (14.3 + 7.17i)4-s + (8.04 + 8.04i)5-s − 49.8·7-s + (−49.0 − 41.0i)8-s + (−23.8 − 38.7i)10-s + (84.2 − 84.2i)11-s + (19.4 − 19.4i)13-s + (194. + 45.9i)14-s + (153. + 205. i)16-s − 437.·17-s + (349. + 349. i)19-s + (57.3 + 172. i)20-s + (−405. + 250. i)22-s + 404.·23-s + ⋯
L(s)  = 1  + (−0.973 − 0.230i)2-s + (0.893 + 0.448i)4-s + (0.321 + 0.321i)5-s − 1.01·7-s + (−0.766 − 0.642i)8-s + (−0.238 − 0.387i)10-s + (0.696 − 0.696i)11-s + (0.115 − 0.115i)13-s + (0.990 + 0.234i)14-s + (0.598 + 0.801i)16-s − 1.51·17-s + (0.966 + 0.966i)19-s + (0.143 + 0.431i)20-s + (−0.838 + 0.517i)22-s + 0.765·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.245 + 0.969i$
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: \((2,\ 144,\ (\ :2),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.716004 - 0.557122i\)
\(L(\frac12)\) \(\approx\) \(0.716004 - 0.557122i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.89 + 0.921i)T \)
3 \( 1 \)
good5 \( 1 + (-8.04 - 8.04i)T + 625iT^{2} \)
7 \( 1 + 49.8T + 2.40e3T^{2} \)
11 \( 1 + (-84.2 + 84.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (-19.4 + 19.4i)T - 2.85e4iT^{2} \)
17 \( 1 + 437.T + 8.35e4T^{2} \)
19 \( 1 + (-349. - 349. i)T + 1.30e5iT^{2} \)
23 \( 1 - 404.T + 2.79e5T^{2} \)
29 \( 1 + (-1.03e3 + 1.03e3i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.50e3iT - 9.23e5T^{2} \)
37 \( 1 + (434. + 434. i)T + 1.87e6iT^{2} \)
41 \( 1 + 696. iT - 2.82e6T^{2} \)
43 \( 1 + (-917. + 917. i)T - 3.41e6iT^{2} \)
47 \( 1 + 111. iT - 4.87e6T^{2} \)
53 \( 1 + (1.04e3 + 1.04e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.71e3 + 1.71e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.71e3 + 3.71e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.85e3 + 1.85e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + 905. iT - 2.83e7T^{2} \)
79 \( 1 - 5.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.56e3 - 7.56e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 6.43e3iT - 6.27e7T^{2} \)
97 \( 1 + 413.T + 8.85e7T^{2} \)
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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

−11.94468599852480034102731814684, −11.03495272033304532485365869075, −9.975866110539957970671201601662, −9.246982195097044735084362552370, −8.158762679730442641653375763145, −6.75009590479787936725479990789, −6.09267464178746762102160876867, −3.72200881947588014582890682563, −2.42252473854482319401342887464, −0.55183708149761486415092245458, 1.24061782965770926847366593455, 2.93794956260707445751958242590, 4.98956537244577579103415345723, 6.57525457355778684972442853660, 7.07052232390400285716309718244, 8.876773057859597757073013955439, 9.245484312019411597380728980500, 10.34568099235966641227870755695, 11.42164467242861116327122801409, 12.52749013705795068419723377224

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