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Properties

Label 2-12e2-16.3-c2-0-17
Degree $2$
Conductor $144$
Sign $0.407 + 0.913i$
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  + (1.84 − 0.777i)2-s + (2.79 − 2.86i)4-s + (4.78 − 4.78i)5-s − 10.3·7-s + (2.91 − 7.45i)8-s + (5.09 − 12.5i)10-s + (0.526 + 0.526i)11-s + (17.2 + 17.2i)13-s + (−19.0 + 8.03i)14-s + (−0.429 − 15.9i)16-s − 4.71·17-s + (−2.53 + 2.53i)19-s + (−0.363 − 27.0i)20-s + (1.37 + 0.560i)22-s + 12.5·23-s + ⋯
L(s)  = 1  + (0.921 − 0.388i)2-s + (0.697 − 0.716i)4-s + (0.957 − 0.957i)5-s − 1.47·7-s + (0.364 − 0.931i)8-s + (0.509 − 1.25i)10-s + (0.0478 + 0.0478i)11-s + (1.32 + 1.32i)13-s + (−1.35 + 0.573i)14-s + (−0.0268 − 0.999i)16-s − 0.277·17-s + (−0.133 + 0.133i)19-s + (−0.0181 − 1.35i)20-s + (0.0626 + 0.0254i)22-s + 0.547·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.407 + 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08789 - 1.35490i\)
\(L(\frac12)\) \(\approx\) \(2.08789 - 1.35490i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.84 + 0.777i)T \)
3 \( 1 \)
good5 \( 1 + (-4.78 + 4.78i)T - 25iT^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + (-0.526 - 0.526i)T + 121iT^{2} \)
13 \( 1 + (-17.2 - 17.2i)T + 169iT^{2} \)
17 \( 1 + 4.71T + 289T^{2} \)
19 \( 1 + (2.53 - 2.53i)T - 361iT^{2} \)
23 \( 1 - 12.5T + 529T^{2} \)
29 \( 1 + (-2.19 - 2.19i)T + 841iT^{2} \)
31 \( 1 - 28.0iT - 961T^{2} \)
37 \( 1 + (32.1 - 32.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 23.1iT - 1.68e3T^{2} \)
43 \( 1 + (-4.79 - 4.79i)T + 1.84e3iT^{2} \)
47 \( 1 - 39.0iT - 2.20e3T^{2} \)
53 \( 1 + (-27.9 + 27.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (79.8 + 79.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (36.7 + 36.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (10.9 - 10.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 52.6T + 5.04e3T^{2} \)
73 \( 1 + 67.8iT - 5.32e3T^{2} \)
79 \( 1 - 56.4iT - 6.24e3T^{2} \)
83 \( 1 + (-58.3 + 58.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 - 60.9T + 9.40e3T^{2} \)
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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

−12.94179651115547139082786588140, −11.97262343709418238167002438604, −10.71432109839992438962756309883, −9.601541250351429518195336110582, −8.926366538447554541486032721824, −6.67192114236798560048218625224, −6.08301328856159202755332833713, −4.73376173253909935509704277018, −3.34961795409989662784877299080, −1.56201327682714092690743358829, 2.68773553083683797449845834260, 3.63409029676305589064330915256, 5.72436985348625344223956305427, 6.26688018533525329656286858164, 7.27054852377895917001840475035, 8.868225093574596633019494116395, 10.25949400174949967727557722488, 10.92376667719909118684520195215, 12.43342773459326130687720028014, 13.36059462285024927532098590908

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