Properties

Degree $2$
Conductor $144$
Sign $-0.943 - 0.330i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 − 0.362i)2-s + (3.73 + 1.42i)4-s + (−1.69 + 1.69i)5-s − 5.74·7-s + (−6.83 − 4.16i)8-s + (3.95 − 2.72i)10-s + (5.59 + 5.59i)11-s + (−13.5 − 13.5i)13-s + (11.2 + 2.08i)14-s + (11.9 + 10.6i)16-s − 19.7·17-s + (−21.6 + 21.6i)19-s + (−8.77 + 3.92i)20-s + (−8.97 − 13.0i)22-s − 24.9·23-s + ⋯
L(s)  = 1  + (−0.983 − 0.181i)2-s + (0.934 + 0.356i)4-s + (−0.339 + 0.339i)5-s − 0.820·7-s + (−0.853 − 0.520i)8-s + (0.395 − 0.272i)10-s + (0.508 + 0.508i)11-s + (−1.04 − 1.04i)13-s + (0.806 + 0.148i)14-s + (0.745 + 0.666i)16-s − 1.15·17-s + (−1.14 + 1.14i)19-s + (−0.438 + 0.196i)20-s + (−0.407 − 0.592i)22-s − 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.943 - 0.330i$
Motivic weight: \(2\)
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.943 - 0.330i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0223078 + 0.131029i\)
\(L(\frac12)\) \(\approx\) \(0.0223078 + 0.131029i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.362i)T \)
3 \( 1 \)
good5 \( 1 + (1.69 - 1.69i)T - 25iT^{2} \)
7 \( 1 + 5.74T + 49T^{2} \)
11 \( 1 + (-5.59 - 5.59i)T + 121iT^{2} \)
13 \( 1 + (13.5 + 13.5i)T + 169iT^{2} \)
17 \( 1 + 19.7T + 289T^{2} \)
19 \( 1 + (21.6 - 21.6i)T - 361iT^{2} \)
23 \( 1 + 24.9T + 529T^{2} \)
29 \( 1 + (1.50 + 1.50i)T + 841iT^{2} \)
31 \( 1 + 2.20iT - 961T^{2} \)
37 \( 1 + (-27.6 + 27.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 51.3iT - 1.68e3T^{2} \)
43 \( 1 + (-21.4 - 21.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 76.5iT - 2.20e3T^{2} \)
53 \( 1 + (-56.5 + 56.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-48.0 - 48.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (51.5 + 51.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (-63.4 + 63.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 43.4T + 5.04e3T^{2} \)
73 \( 1 - 73.9iT - 5.32e3T^{2} \)
79 \( 1 + 4.12iT - 6.24e3T^{2} \)
83 \( 1 + (38.4 - 38.4i)T - 6.88e3iT^{2} \)
89 \( 1 - 52.9iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−12.85583584980199403592705276027, −12.28698506021886136647534260942, −11.06326115498927314309107388635, −10.14396049609647729558315105515, −9.352091793159606088000350683017, −8.077811994296712131146298977132, −7.11190797096484576127138253823, −6.06100646258279122653551904143, −3.87236494851724605967228919680, −2.34602497499229229261589002136, 0.10825706632679258635360455875, 2.39451353829254491026770772904, 4.38231872036453389023391747881, 6.28739131984038355139740053337, 6.98708694077748555200481699386, 8.443031607755895994028158875497, 9.181010522231483353929967235851, 10.14638288945512874098201425869, 11.36806969529709063782543683367, 12.11683852784730572812734729945

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