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Properties

Label 2-12e2-48.5-c2-0-12
Degree $2$
Conductor $144$
Sign $0.999 - 0.00727i$
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 + 0.107i)2-s + (3.97 + 0.430i)4-s + (1.92 − 1.92i)5-s + 2.43i·7-s + (7.89 + 1.28i)8-s + (4.05 − 3.64i)10-s + (2.79 − 2.79i)11-s + (−1.54 + 1.54i)13-s + (−0.261 + 4.85i)14-s + (15.6 + 3.42i)16-s − 20.3i·17-s + (−25.0 + 25.0i)19-s + (8.49 − 6.83i)20-s + (5.87 − 5.27i)22-s + 3.55·23-s + ⋯
L(s)  = 1  + (0.998 + 0.0538i)2-s + (0.994 + 0.107i)4-s + (0.385 − 0.385i)5-s + 0.347i·7-s + (0.986 + 0.160i)8-s + (0.405 − 0.364i)10-s + (0.253 − 0.253i)11-s + (−0.118 + 0.118i)13-s + (−0.0187 + 0.346i)14-s + (0.976 + 0.213i)16-s − 1.19i·17-s + (−1.31 + 1.31i)19-s + (0.424 − 0.341i)20-s + (0.267 − 0.239i)22-s + 0.154·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00727i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.00727i$
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: \((2,\ 144,\ (\ :1),\ 0.999 - 0.00727i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.68709 + 0.00978101i\)
\(L(\frac12)\) \(\approx\) \(2.68709 + 0.00978101i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.107i)T \)
3 \( 1 \)
good5 \( 1 + (-1.92 + 1.92i)T - 25iT^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 + (-2.79 + 2.79i)T - 121iT^{2} \)
13 \( 1 + (1.54 - 1.54i)T - 169iT^{2} \)
17 \( 1 + 20.3iT - 289T^{2} \)
19 \( 1 + (25.0 - 25.0i)T - 361iT^{2} \)
23 \( 1 - 3.55T + 529T^{2} \)
29 \( 1 + (23.4 + 23.4i)T + 841iT^{2} \)
31 \( 1 + 13.3T + 961T^{2} \)
37 \( 1 + (25.4 + 25.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 64.0T + 1.68e3T^{2} \)
43 \( 1 + (24.6 + 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 - 79.5iT - 2.20e3T^{2} \)
53 \( 1 + (-39.8 + 39.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (19.2 - 19.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (-63.5 + 63.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (65.1 - 65.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 84.6T + 5.04e3T^{2} \)
73 \( 1 - 39.7iT - 5.32e3T^{2} \)
79 \( 1 - 109.T + 6.24e3T^{2} \)
83 \( 1 + (14.7 + 14.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 32.3T + 7.92e3T^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−12.91205089504847924374279142406, −12.06288636618266099024035362573, −11.13144830093318269163312433742, −9.902584820561384057858645158910, −8.630748088132888741666556978847, −7.28039484064614748041238047080, −6.05062966515665189987185042742, −5.10550456679958826019634879453, −3.70415161804512369926325774680, −2.02392218141616396826376997749, 2.07333577741298954782937650991, 3.66448030201510494690348512322, 4.94100999318586439875754736890, 6.32247390017525705216785351177, 7.08572238902193178757630359643, 8.605220561214419236784710568290, 10.24044581289191401757366829008, 10.83529854084224311721508107209, 12.03378934512644449463965318172, 13.02941945275351195055063814787

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