Properties

Degree $2$
Conductor $144$
Sign $-0.176 + 0.984i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 − 0.863i)2-s + (2.50 − 3.11i)4-s + (−6.49 − 6.49i)5-s + 3.94·7-s + (1.83 − 7.78i)8-s + (−17.3 − 6.10i)10-s + (−4.31 + 4.31i)11-s + (4.06 − 4.06i)13-s + (7.11 − 3.40i)14-s + (−3.41 − 15.6i)16-s + 14.5·17-s + (4.94 + 4.94i)19-s + (−36.5 + 3.94i)20-s + (−4.05 + 11.4i)22-s + 43.6·23-s + ⋯
L(s)  = 1  + (0.901 − 0.431i)2-s + (0.627 − 0.778i)4-s + (−1.29 − 1.29i)5-s + 0.563·7-s + (0.229 − 0.973i)8-s + (−1.73 − 0.610i)10-s + (−0.391 + 0.391i)11-s + (0.312 − 0.312i)13-s + (0.508 − 0.243i)14-s + (−0.213 − 0.976i)16-s + 0.856·17-s + (0.260 + 0.260i)19-s + (−1.82 + 0.197i)20-s + (−0.184 + 0.522i)22-s + 1.89·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.176 + 0.984i$
Motivic weight: \(2\)
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.176 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29824 - 1.55221i\)
\(L(\frac12)\) \(\approx\) \(1.29824 - 1.55221i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.80 + 0.863i)T \)
3 \( 1 \)
good5 \( 1 + (6.49 + 6.49i)T + 25iT^{2} \)
7 \( 1 - 3.94T + 49T^{2} \)
11 \( 1 + (4.31 - 4.31i)T - 121iT^{2} \)
13 \( 1 + (-4.06 + 4.06i)T - 169iT^{2} \)
17 \( 1 - 14.5T + 289T^{2} \)
19 \( 1 + (-4.94 - 4.94i)T + 361iT^{2} \)
23 \( 1 - 43.6T + 529T^{2} \)
29 \( 1 + (25.0 - 25.0i)T - 841iT^{2} \)
31 \( 1 + 32.5iT - 961T^{2} \)
37 \( 1 + (-4.14 - 4.14i)T + 1.36e3iT^{2} \)
41 \( 1 - 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (16.1 - 16.1i)T - 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (-31.5 - 31.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (-49.7 + 49.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (-44.4 + 44.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (1.64 + 1.64i)T + 4.48e3iT^{2} \)
71 \( 1 + 24.1T + 5.04e3T^{2} \)
73 \( 1 + 10.7iT - 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (42.0 + 42.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−12.70937385601935643661129583149, −11.66889169090886047493642138432, −11.05179960561132629774143024926, −9.562379484710145900426748758712, −8.242833841376511148983524224962, −7.30299687571664515306954695307, −5.38916051242331350707671932014, −4.65891237354220929603585914839, −3.42071960558415044116746557829, −1.14522633961299544255791614652, 2.91589639941264496792319099524, 3.90960802614772315777528798469, 5.34322619330605735318086113948, 6.85163669563923634860585531509, 7.51463156979200030998506258935, 8.519259695691493863263546248379, 10.63295757288180843525660479386, 11.31098793082894464473584517305, 12.00567690870570099632569795960, 13.29594136384567130765119880378

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