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Properties

Label 2-12e2-48.5-c2-0-9
Degree $2$
Conductor $144$
Sign $0.972 + 0.231i$
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  + (−1.11 + 1.65i)2-s + (−1.51 − 3.70i)4-s + (−0.900 + 0.900i)5-s − 7.66i·7-s + (7.83 + 1.62i)8-s + (−0.489 − 2.49i)10-s + (4.57 − 4.57i)11-s + (10.9 − 10.9i)13-s + (12.7 + 8.55i)14-s + (−11.4 + 11.1i)16-s + 0.0435i·17-s + (12.9 − 12.9i)19-s + (4.69 + 1.97i)20-s + (2.48 + 12.7i)22-s + 18.3·23-s + ⋯
L(s)  = 1  + (−0.557 + 0.829i)2-s + (−0.377 − 0.925i)4-s + (−0.180 + 0.180i)5-s − 1.09i·7-s + (0.979 + 0.203i)8-s + (−0.0489 − 0.249i)10-s + (0.415 − 0.415i)11-s + (0.844 − 0.844i)13-s + (0.908 + 0.610i)14-s + (−0.714 + 0.699i)16-s + 0.00256i·17-s + (0.683 − 0.683i)19-s + (0.234 + 0.0987i)20-s + (0.113 + 0.577i)22-s + 0.796·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.972 + 0.231i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.972 + 0.231i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.997592 - 0.117179i\)
\(L(\frac12)\) \(\approx\) \(0.997592 - 0.117179i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 1.65i)T \)
3 \( 1 \)
good5 \( 1 + (0.900 - 0.900i)T - 25iT^{2} \)
7 \( 1 + 7.66iT - 49T^{2} \)
11 \( 1 + (-4.57 + 4.57i)T - 121iT^{2} \)
13 \( 1 + (-10.9 + 10.9i)T - 169iT^{2} \)
17 \( 1 - 0.0435iT - 289T^{2} \)
19 \( 1 + (-12.9 + 12.9i)T - 361iT^{2} \)
23 \( 1 - 18.3T + 529T^{2} \)
29 \( 1 + (34.9 + 34.9i)T + 841iT^{2} \)
31 \( 1 - 38.4T + 961T^{2} \)
37 \( 1 + (-11.3 - 11.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 45.6T + 1.68e3T^{2} \)
43 \( 1 + (51.4 + 51.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 56.5iT - 2.20e3T^{2} \)
53 \( 1 + (44.6 - 44.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (20.7 - 20.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (2.42 - 2.42i)T - 3.72e3iT^{2} \)
67 \( 1 + (18.9 - 18.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 114.T + 5.04e3T^{2} \)
73 \( 1 - 127. iT - 5.32e3T^{2} \)
79 \( 1 - 37.0T + 6.24e3T^{2} \)
83 \( 1 + (-84.4 - 84.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 136.T + 7.92e3T^{2} \)
97 \( 1 + 173.T + 9.40e3T^{2} \)
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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

−13.44194847018851374197750719426, −11.46360458406140197682083225570, −10.65632625421269546789258997204, −9.653545669320556995311614183487, −8.475752737947728388827271230227, −7.48360932883363629626226078421, −6.55913349056660597459722865563, −5.24100356134273084886601396647, −3.68860643283984959145355343575, −0.893036658580454510162966466009, 1.67533227139605671150181000096, 3.32706430805126596793075895863, 4.80076385982580350389579834788, 6.49795240289627637307175631472, 7.997274753612816993393832859943, 8.943618014519326308059076157611, 9.667637204279215155572584101182, 11.02334564363827450302518570428, 11.86249852463320117198154411593, 12.52646114390943562291356006877

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