Properties

Label 2-12e2-144.5-c2-0-13
Degree $2$
Conductor $144$
Sign $0.274 - 0.961i$
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  + (−0.786 + 1.83i)2-s + (1.51 − 2.58i)3-s + (−2.76 − 2.89i)4-s + (1.99 + 7.43i)5-s + (3.56 + 4.82i)6-s + (−1.50 + 0.869i)7-s + (7.48 − 2.81i)8-s + (−4.38 − 7.86i)9-s + (−15.2 − 2.17i)10-s + (15.2 + 4.09i)11-s + (−11.6 + 2.75i)12-s + (5.73 + 21.4i)13-s + (−0.415 − 3.45i)14-s + (22.2 + 6.14i)15-s + (−0.717 + 15.9i)16-s + 1.12i·17-s + ⋯
L(s)  = 1  + (−0.393 + 0.919i)2-s + (0.506 − 0.862i)3-s + (−0.691 − 0.722i)4-s + (0.398 + 1.48i)5-s + (0.593 + 0.804i)6-s + (−0.215 + 0.124i)7-s + (0.936 − 0.351i)8-s + (−0.486 − 0.873i)9-s + (−1.52 − 0.217i)10-s + (1.39 + 0.372i)11-s + (−0.973 + 0.229i)12-s + (0.441 + 1.64i)13-s + (−0.0296 − 0.246i)14-s + (1.48 + 0.409i)15-s + (−0.0448 + 0.998i)16-s + 0.0664i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.274 - 0.961i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.274 - 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.11639 + 0.842188i\)
\(L(\frac12)\) \(\approx\) \(1.11639 + 0.842188i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 - 1.83i)T \)
3 \( 1 + (-1.51 + 2.58i)T \)
good5 \( 1 + (-1.99 - 7.43i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (1.50 - 0.869i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-15.2 - 4.09i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (-5.73 - 21.4i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 - 1.12iT - 289T^{2} \)
19 \( 1 + (-12.2 + 12.2i)T - 361iT^{2} \)
23 \( 1 + (6.80 - 11.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.249 + 0.932i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (-1.87 + 3.24i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (35.9 + 35.9i)T + 1.36e3iT^{2} \)
41 \( 1 + (-30.2 + 52.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-4.42 + 16.5i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-11.2 + 6.47i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.2 + 38.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (-5.85 - 21.8i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (95.2 + 25.5i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (20.8 + 77.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 96.9T + 5.04e3T^{2} \)
73 \( 1 + 19.4iT - 5.32e3T^{2} \)
79 \( 1 + (60.9 + 105. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (21.7 - 81.1i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 62.3T + 7.92e3T^{2} \)
97 \( 1 + (-63.8 - 110. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.74109797366995170042092686143, −12.06916768459455726035608818136, −10.98805134873506766115507907291, −9.504383284028225242373565059455, −8.995004416901540322974138683819, −7.34251284188926736005370159894, −6.79993612599589938946162114280, −6.07530849918146202287387106424, −3.79141397008769405980887498775, −1.86128941659285366865020158949, 1.17242047984302669981535538788, 3.24997316783275446351186554223, 4.39976052582400136620513466585, 5.60249670105620597197644898380, 8.080492111488947978989397523445, 8.732647109136586921185243287988, 9.586508929542731427263854526329, 10.34363955256764093224742580273, 11.61125259230284446641085763643, 12.62825180852826501020336311595

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