Properties

Label 2-12e2-144.5-c2-0-26
Degree $2$
Conductor $144$
Sign $-0.646 + 0.762i$
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  + (−0.762 − 1.84i)2-s + (−2.07 + 2.16i)3-s + (−2.83 + 2.81i)4-s + (−0.115 − 0.432i)5-s + (5.58 + 2.17i)6-s + (7.23 − 4.17i)7-s + (7.37 + 3.09i)8-s + (−0.405 − 8.99i)9-s + (−0.710 + 0.543i)10-s + (−20.2 − 5.42i)11-s + (−0.230 − 11.9i)12-s + (−1.29 − 4.82i)13-s + (−13.2 − 10.1i)14-s + (1.17 + 0.644i)15-s + (0.104 − 15.9i)16-s − 25.9i·17-s + ⋯
L(s)  = 1  + (−0.381 − 0.924i)2-s + (−0.691 + 0.722i)3-s + (−0.709 + 0.704i)4-s + (−0.0231 − 0.0864i)5-s + (0.931 + 0.363i)6-s + (1.03 − 0.596i)7-s + (0.921 + 0.387i)8-s + (−0.0450 − 0.998i)9-s + (−0.0710 + 0.0543i)10-s + (−1.83 − 0.492i)11-s + (−0.0192 − 0.999i)12-s + (−0.0995 − 0.371i)13-s + (−0.945 − 0.728i)14-s + (0.0784 + 0.0429i)15-s + (0.00656 − 0.999i)16-s − 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.646 + 0.762i$
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: \((2,\ 144,\ (\ :1),\ -0.646 + 0.762i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269291 - 0.581415i\)
\(L(\frac12)\) \(\approx\) \(0.269291 - 0.581415i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.762 + 1.84i)T \)
3 \( 1 + (2.07 - 2.16i)T \)
good5 \( 1 + (0.115 + 0.432i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (-7.23 + 4.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (20.2 + 5.42i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (1.29 + 4.82i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + 25.9iT - 289T^{2} \)
19 \( 1 + (-1.80 + 1.80i)T - 361iT^{2} \)
23 \( 1 + (1.78 - 3.10i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.2 + 49.2i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (14.2 - 24.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (22.5 + 22.5i)T + 1.36e3iT^{2} \)
41 \( 1 + (16.3 - 28.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (4.06 - 15.1i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-33.7 + 19.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (57.3 - 57.3i)T - 2.80e3iT^{2} \)
59 \( 1 + (11.5 + 43.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-70.8 - 18.9i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-12.2 - 45.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 26.4T + 5.04e3T^{2} \)
73 \( 1 + 38.6iT - 5.32e3T^{2} \)
79 \( 1 + (29.8 + 51.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (12.5 - 46.7i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 79.3T + 7.92e3T^{2} \)
97 \( 1 + (16.3 + 28.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.16851408886735942357476191487, −11.19847828038385073274614278022, −10.63250006600248868746698720880, −9.790961683729328433247096111831, −8.452828881686740214840270939162, −7.46914662880404948199098424241, −5.28844537092097263088685073882, −4.57497507172176790255655824791, −2.89469743963364184236171930427, −0.52352914918075591517175619357, 1.82690814527747409005020100976, 4.90972616777037426795249912563, 5.55276949142004776707842529065, 6.90024582183627261304536106383, 7.889520844862919531000587600254, 8.594673994673203134718982536424, 10.34307094635289952339823319662, 10.97698778034101423491323191237, 12.44724472745573614008443854998, 13.15833815202538435704920351215

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