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Properties

Label 2-12e2-144.11-c1-0-19
Degree $2$
Conductor $144$
Sign $-0.195 + 0.980i$
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  + (0.772 − 1.18i)2-s + (−0.501 − 1.65i)3-s + (−0.806 − 1.83i)4-s + (3.72 + 0.997i)5-s + (−2.35 − 0.686i)6-s + (−0.481 + 0.833i)7-s + (−2.79 − 0.458i)8-s + (−2.49 + 1.66i)9-s + (4.05 − 3.63i)10-s + (−3.75 + 1.00i)11-s + (−2.63 + 2.25i)12-s + (1.60 + 0.430i)13-s + (0.615 + 1.21i)14-s + (−0.213 − 6.67i)15-s + (−2.69 + 2.95i)16-s − 2.58i·17-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.289 − 0.957i)3-s + (−0.403 − 0.915i)4-s + (1.66 + 0.445i)5-s + (−0.959 − 0.280i)6-s + (−0.181 + 0.315i)7-s + (−0.986 − 0.162i)8-s + (−0.832 + 0.554i)9-s + (1.28 − 1.15i)10-s + (−1.13 + 0.303i)11-s + (−0.759 + 0.650i)12-s + (0.445 + 0.119i)13-s + (0.164 + 0.324i)14-s + (−0.0550 − 1.72i)15-s + (−0.674 + 0.737i)16-s − 0.626i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920381 - 1.12230i\)
\(L(\frac12)\) \(\approx\) \(0.920381 - 1.12230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.772 + 1.18i)T \)
3 \( 1 + (0.501 + 1.65i)T \)
good5 \( 1 + (-3.72 - 0.997i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.481 - 0.833i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.75 - 1.00i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.60 - 0.430i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + (-4.02 - 4.02i)T + 19iT^{2} \)
23 \( 1 + (-0.600 + 0.346i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.75 - 1.54i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.07 + 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.11 + 2.11i)T + 37iT^{2} \)
41 \( 1 + (4.97 + 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.78 + 6.65i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.88 + 5.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.68 - 6.68i)T - 53iT^{2} \)
59 \( 1 + (2.34 - 8.75i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.570 - 2.12i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.12 - 4.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.40iT - 71T^{2} \)
73 \( 1 - 3.77iT - 73T^{2} \)
79 \( 1 + (9.52 + 5.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.35 - 8.79i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (-1.76 + 3.05i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.89932795426023321317819975221, −12.02621704835195063665239999443, −10.80347705199381608330653387316, −10.05261388621512447599841345736, −8.957542870987830062667891581711, −7.16932068793567240871378103443, −5.84008247878780132358284313377, −5.40127453196596880992614681582, −2.86245343897908834045125199154, −1.80729443139721721712468793153, 3.08905815284900212354190935436, 4.83817311762816135082586879129, 5.56481835683032832768090832427, 6.47465148595887808778483862833, 8.233354889599067213524984140931, 9.322794560275619298498286256804, 10.09243473545403511227797959663, 11.29761805455348560807815487065, 12.91129736705602889663788464338, 13.43052989537163918331060539985

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