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Properties

Label 2-12e2-144.85-c1-0-8
Degree $2$
Conductor $144$
Sign $0.888 + 0.459i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.751 + 1.19i)2-s + (−1.64 − 0.543i)3-s + (−0.871 − 1.79i)4-s + (−1.60 + 0.430i)5-s + (1.88 − 1.56i)6-s + (3.62 − 2.09i)7-s + (2.81 + 0.307i)8-s + (2.40 + 1.78i)9-s + (0.690 − 2.24i)10-s + (1.24 − 4.63i)11-s + (0.456 + 3.43i)12-s + (−0.879 − 3.28i)13-s + (−0.214 + 5.91i)14-s + (2.87 + 0.164i)15-s + (−2.47 + 3.13i)16-s − 2.14·17-s + ⋯
L(s)  = 1  + (−0.531 + 0.847i)2-s + (−0.949 − 0.313i)3-s + (−0.435 − 0.899i)4-s + (−0.718 + 0.192i)5-s + (0.770 − 0.637i)6-s + (1.37 − 0.791i)7-s + (0.994 + 0.108i)8-s + (0.803 + 0.595i)9-s + (0.218 − 0.710i)10-s + (0.374 − 1.39i)11-s + (0.131 + 0.991i)12-s + (−0.243 − 0.910i)13-s + (−0.0573 + 1.58i)14-s + (0.742 + 0.0425i)15-s + (−0.619 + 0.784i)16-s − 0.519·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.888 + 0.459i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.888 + 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577955 - 0.140736i\)
\(L(\frac12)\) \(\approx\) \(0.577955 - 0.140736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.751 - 1.19i)T \)
3 \( 1 + (1.64 + 0.543i)T \)
good5 \( 1 + (1.60 - 0.430i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.62 + 2.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.24 + 4.63i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.879 + 3.28i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + (-1.03 + 1.03i)T - 19iT^{2} \)
23 \( 1 + (0.405 + 0.234i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.55 - 1.75i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.18 + 5.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.728 + 0.728i)T + 37iT^{2} \)
41 \( 1 + (-2.52 - 1.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.84 - 10.6i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.61 + 7.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.17 - 1.17i)T + 53iT^{2} \)
59 \( 1 + (1.48 - 0.397i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.53 - 2.01i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.63 - 9.82i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.27iT - 71T^{2} \)
73 \( 1 - 8.16iT - 73T^{2} \)
79 \( 1 + (-3.63 - 6.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.59 + 2.03i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (5.67 + 9.83i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.25116339999513870621846362532, −11.52598103801883712412094573988, −11.12584930445176598107278096793, −10.13867696046455411027558364546, −8.331041040782975314754571806741, −7.77229763329756335913795739026, −6.67411200012382668300977907815, −5.44714354565001563554138651014, −4.33609608274436373264574318744, −0.888328255464630102370073011805, 1.82348209257845144311920295138, 4.27234856474894710218587008940, 4.88659717405230109152700167737, 6.92814223192410141902767024465, 8.127353791118771055064552103875, 9.214391151465988874135848914016, 10.27618512046854265637493435439, 11.41167624570437468624135345329, 11.95716647920998398214472275253, 12.40555201298084289272755514318

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