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Properties

Label 2-12e2-9.2-c2-0-0
Degree $2$
Conductor $144$
Sign $-0.898 - 0.439i$
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  + (0.419 − 2.97i)3-s + (−8.20 + 4.73i)5-s + (−1.05 + 1.83i)7-s + (−8.64 − 2.49i)9-s + (−13.7 − 7.91i)11-s + (4.70 + 8.14i)13-s + (10.6 + 26.3i)15-s + 11.6i·17-s − 12.9·19-s + (4.99 + 3.90i)21-s + (5.27 − 3.04i)23-s + (32.4 − 56.1i)25-s + (−11.0 + 24.6i)27-s + (−24.7 − 14.2i)29-s + (−8.75 − 15.1i)31-s + ⋯
L(s)  = 1  + (0.139 − 0.990i)3-s + (−1.64 + 0.947i)5-s + (−0.150 + 0.261i)7-s + (−0.960 − 0.276i)9-s + (−1.24 − 0.719i)11-s + (0.361 + 0.626i)13-s + (0.708 + 1.75i)15-s + 0.682i·17-s − 0.682·19-s + (0.237 + 0.186i)21-s + (0.229 − 0.132i)23-s + (1.29 − 2.24i)25-s + (−0.408 + 0.912i)27-s + (−0.854 − 0.493i)29-s + (−0.282 − 0.489i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.898 - 0.439i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00767223 + 0.0331310i\)
\(L(\frac12)\) \(\approx\) \(0.00767223 + 0.0331310i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.419 + 2.97i)T \)
good5 \( 1 + (8.20 - 4.73i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.05 - 1.83i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (13.7 + 7.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.70 - 8.14i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 11.6iT - 289T^{2} \)
19 \( 1 + 12.9T + 361T^{2} \)
23 \( 1 + (-5.27 + 3.04i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.7 + 14.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (8.75 + 15.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 15.6T + 1.36e3T^{2} \)
41 \( 1 + (-14.8 + 8.54i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (20.6 + 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + (38.5 - 22.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.86 - 3.22i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.0 + 36.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 120. iT - 5.04e3T^{2} \)
73 \( 1 - 5.48T + 5.32e3T^{2} \)
79 \( 1 + (60.5 - 104. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (46.5 + 26.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 102. iT - 7.92e3T^{2} \)
97 \( 1 + (58.9 - 102. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.16412197044152952987730156324, −12.26628065899441645717686806209, −11.29427574471257982832403504227, −10.70661166423332070062437340936, −8.688695857375195106935756573450, −7.918394544196283837294000447402, −7.08033796819116596575733172657, −5.93634907568890801650644603800, −3.87673321293001182425882626827, −2.63823956140511072312027052039, 0.02082722439602907141299993727, 3.26547724910180908429594352918, 4.44308875797039482201019158037, 5.25758312208750005943862217554, 7.44928055078729426842713606696, 8.239847199079663694699549719078, 9.225477477908854952163289689266, 10.50862576661076132350179104839, 11.29197469027511974714322829136, 12.43487970577712356461064889480

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