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Properties

Label 2-12e2-16.11-c2-0-6
Degree $2$
Conductor $144$
Sign $0.992 + 0.118i$
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  + (−1.99 + 0.136i)2-s + (3.96 − 0.545i)4-s + (−0.227 − 0.227i)5-s + 3.90·7-s + (−7.83 + 1.63i)8-s + (0.485 + 0.423i)10-s + (2.21 − 2.21i)11-s + (5.08 − 5.08i)13-s + (−7.78 + 0.533i)14-s + (15.4 − 4.32i)16-s + 18.8·17-s + (11.7 + 11.7i)19-s + (−1.02 − 0.777i)20-s + (−4.10 + 4.71i)22-s + 35.4·23-s + ⋯
L(s)  = 1  + (−0.997 + 0.0683i)2-s + (0.990 − 0.136i)4-s + (−0.0455 − 0.0455i)5-s + 0.557·7-s + (−0.978 + 0.203i)8-s + (0.0485 + 0.0423i)10-s + (0.200 − 0.200i)11-s + (0.391 − 0.391i)13-s + (−0.556 + 0.0381i)14-s + (0.962 − 0.270i)16-s + 1.10·17-s + (0.619 + 0.619i)19-s + (−0.0513 − 0.0388i)20-s + (−0.186 + 0.214i)22-s + 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.992 + 0.118i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.992 + 0.118i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02951 - 0.0613093i\)
\(L(\frac12)\) \(\approx\) \(1.02951 - 0.0613093i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.136i)T \)
3 \( 1 \)
good5 \( 1 + (0.227 + 0.227i)T + 25iT^{2} \)
7 \( 1 - 3.90T + 49T^{2} \)
11 \( 1 + (-2.21 + 2.21i)T - 121iT^{2} \)
13 \( 1 + (-5.08 + 5.08i)T - 169iT^{2} \)
17 \( 1 - 18.8T + 289T^{2} \)
19 \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \)
23 \( 1 - 35.4T + 529T^{2} \)
29 \( 1 + (-21.2 + 21.2i)T - 841iT^{2} \)
31 \( 1 - 35.9iT - 961T^{2} \)
37 \( 1 + (34.4 + 34.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.1iT - 1.68e3T^{2} \)
43 \( 1 + (28.3 - 28.3i)T - 1.84e3iT^{2} \)
47 \( 1 + 32.8iT - 2.20e3T^{2} \)
53 \( 1 + (42.1 + 42.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (66.9 - 66.9i)T - 3.48e3iT^{2} \)
61 \( 1 + (-17.3 + 17.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-39.6 - 39.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 63.0T + 5.04e3T^{2} \)
73 \( 1 + 75.4iT - 5.32e3T^{2} \)
79 \( 1 - 59.1iT - 6.24e3T^{2} \)
83 \( 1 + (71.2 + 71.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 150. iT - 7.92e3T^{2} \)
97 \( 1 - 51.5T + 9.40e3T^{2} \)
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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

−12.53531865077769118379763494246, −11.64366653873283045615454028813, −10.67596255396592777922583515485, −9.764397042685093323255127920749, −8.554174742114609838938915156872, −7.81326636517261480315674427501, −6.55763217112061933691052189799, −5.25322277311078344006708694906, −3.17255189651736122924299696418, −1.22217191926814477972654271276, 1.35836975878111873701346469100, 3.22029629802147597822142497253, 5.20758006080305191340531935332, 6.74052419107415514629354112072, 7.67818699796338834482032026497, 8.800986647456238869275556992754, 9.672688934682138261410052448880, 10.87443658150195382330176254777, 11.57025533578416519349270817173, 12.60751352223842414508057684865

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