Properties

Label 2-12e2-9.2-c4-0-21
Degree $2$
Conductor $144$
Sign $-0.993 - 0.117i$
Analytic cond. $14.8852$
Root an. cond. $3.85814$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.256 − 8.99i)3-s + (−7.67 + 4.43i)5-s + (30.9 − 53.5i)7-s + (−80.8 + 4.61i)9-s + (−94.7 − 54.7i)11-s + (77.8 + 134. i)13-s + (41.8 + 67.9i)15-s − 395. i·17-s − 140.·19-s + (−490. − 264. i)21-s + (−802. + 463. i)23-s + (−273. + 473. i)25-s + (62.3 + 726. i)27-s + (323. + 186. i)29-s + (−521. − 903. i)31-s + ⋯
L(s)  = 1  + (−0.0285 − 0.999i)3-s + (−0.307 + 0.177i)5-s + (0.631 − 1.09i)7-s + (−0.998 + 0.0570i)9-s + (−0.783 − 0.452i)11-s + (0.460 + 0.798i)13-s + (0.186 + 0.302i)15-s − 1.37i·17-s − 0.388·19-s + (−1.11 − 0.599i)21-s + (−1.51 + 0.876i)23-s + (−0.437 + 0.757i)25-s + (0.0854 + 0.996i)27-s + (0.385 + 0.222i)29-s + (−0.543 − 0.940i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.993 - 0.117i$
Analytic conductor: \(14.8852\)
Root analytic conductor: \(3.85814\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :2),\ -0.993 - 0.117i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0456467 + 0.776187i\)
\(L(\frac12)\) \(\approx\) \(0.0456467 + 0.776187i\)
\(L(3)\) 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 \( 1 + (0.256 + 8.99i)T \)
good5 \( 1 + (7.67 - 4.43i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-30.9 + 53.5i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (94.7 + 54.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-77.8 - 134. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 395. iT - 8.35e4T^{2} \)
19 \( 1 + 140.T + 1.30e5T^{2} \)
23 \( 1 + (802. - 463. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-323. - 186. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (521. + 903. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 194.T + 1.87e6T^{2} \)
41 \( 1 + (2.34e3 - 1.35e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-167. + 290. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.46e3 + 1.42e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.76e3iT - 7.89e6T^{2} \)
59 \( 1 + (-4.35e3 + 2.51e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.52e3 + 6.10e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.43e3 - 5.95e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 821. iT - 2.54e7T^{2} \)
73 \( 1 - 4.09e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.78e3 + 6.55e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.77e3 + 3.91e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.28e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.89e3 + 3.27e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−11.54535364669021863274164218376, −11.33982045960760308852510879578, −9.886057284464748901497006128146, −8.335880949443813898380263620685, −7.58957005400868817830197543383, −6.67762363289373656929570494178, −5.22722802028549567197152130699, −3.62378944663441614405976398267, −1.86068707783967183375034801165, −0.29660334446422853351632516589, 2.31437406652824639884788343315, 3.92617678284388735586544918010, 5.13065006412048629610695609082, 6.06978843553913670318973038708, 8.260814458409116185154368344883, 8.466863771616648922255020343472, 10.07001197812099489496066199480, 10.70477186722756967029848882996, 11.92686865061517317750136743834, 12.67064624961624332863835065143

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