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Properties

Label 2-12e2-144.11-c1-0-16
Degree 22
Conductor 144144
Sign 0.768+0.639i0.768 + 0.639i
Analytic cond. 1.149841.14984
Root an. cond. 1.072301.07230
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.688 − 1.23i)2-s + (1.10 + 1.33i)3-s + (−1.05 − 1.70i)4-s + (0.664 + 0.178i)5-s + (2.40 − 0.446i)6-s + (0.645 − 1.11i)7-s + (−2.82 + 0.129i)8-s + (−0.559 + 2.94i)9-s + (0.677 − 0.698i)10-s + (3.21 − 0.860i)11-s + (1.10 − 3.28i)12-s + (−4.74 − 1.27i)13-s + (−0.937 − 1.56i)14-s + (0.496 + 1.08i)15-s + (−1.78 + 3.57i)16-s + 5.58i·17-s + ⋯
L(s)  = 1  + (0.486 − 0.873i)2-s + (0.637 + 0.770i)3-s + (−0.526 − 0.850i)4-s + (0.297 + 0.0796i)5-s + (0.983 − 0.182i)6-s + (0.244 − 0.422i)7-s + (−0.998 + 0.0456i)8-s + (−0.186 + 0.982i)9-s + (0.214 − 0.220i)10-s + (0.968 − 0.259i)11-s + (0.319 − 0.947i)12-s + (−1.31 − 0.352i)13-s + (−0.250 − 0.418i)14-s + (0.128 + 0.279i)15-s + (−0.446 + 0.894i)16-s + 1.35i·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.768+0.639i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((0.768+0.639i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.768+0.639i0.768 + 0.639i
Analytic conductor: 1.149841.14984
Root analytic conductor: 1.072301.07230
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ144(11,⋅)\chi_{144} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1/2), 0.768+0.639i)(2,\ 144,\ (\ :1/2),\ 0.768 + 0.639i)

Particular Values

L(1)L(1) ≈\approx 1.49754−0.541483i1.49754 - 0.541483i
L(12)L(\frac12) ≈\approx 1.49754−0.541483i1.49754 - 0.541483i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.688+1.23i)T 1 + (-0.688 + 1.23i)T
3 1+(−1.10−1.33i)T 1 + (-1.10 - 1.33i)T
good5 1+(−0.664−0.178i)T+(4.33+2.5i)T2 1 + (-0.664 - 0.178i)T + (4.33 + 2.5i)T^{2}
7 1+(−0.645+1.11i)T+(−3.5−6.06i)T2 1 + (-0.645 + 1.11i)T + (-3.5 - 6.06i)T^{2}
11 1+(−3.21+0.860i)T+(9.52−5.5i)T2 1 + (-3.21 + 0.860i)T + (9.52 - 5.5i)T^{2}
13 1+(4.74+1.27i)T+(11.2+6.5i)T2 1 + (4.74 + 1.27i)T + (11.2 + 6.5i)T^{2}
17 1−5.58iT−17T2 1 - 5.58iT - 17T^{2}
19 1+(2.49+2.49i)T+19iT2 1 + (2.49 + 2.49i)T + 19iT^{2}
23 1+(2.36−1.36i)T+(11.5−19.9i)T2 1 + (2.36 - 1.36i)T + (11.5 - 19.9i)T^{2}
29 1+(2.95−0.792i)T+(25.1−14.5i)T2 1 + (2.95 - 0.792i)T + (25.1 - 14.5i)T^{2}
31 1+(−5.28+3.04i)T+(15.5−26.8i)T2 1 + (-5.28 + 3.04i)T + (15.5 - 26.8i)T^{2}
37 1+(−0.507−0.507i)T+37iT2 1 + (-0.507 - 0.507i)T + 37iT^{2}
41 1+(−4.89−8.48i)T+(−20.5+35.5i)T2 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.254−0.949i)T+(−37.2+21.5i)T2 1 + (-0.254 - 0.949i)T + (-37.2 + 21.5i)T^{2}
47 1+(−6.13+10.6i)T+(−23.5−40.7i)T2 1 + (-6.13 + 10.6i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.601−0.601i)T−53iT2 1 + (0.601 - 0.601i)T - 53iT^{2}
59 1+(−1.28+4.77i)T+(−51.0−29.5i)T2 1 + (-1.28 + 4.77i)T + (-51.0 - 29.5i)T^{2}
61 1+(2.90+10.8i)T+(−52.8+30.5i)T2 1 + (2.90 + 10.8i)T + (-52.8 + 30.5i)T^{2}
67 1+(0.0295−0.110i)T+(−58.0−33.5i)T2 1 + (0.0295 - 0.110i)T + (-58.0 - 33.5i)T^{2}
71 1+0.0447iT−71T2 1 + 0.0447iT - 71T^{2}
73 1+13.2iT−73T2 1 + 13.2iT - 73T^{2}
79 1+(−2.50−1.44i)T+(39.5+68.4i)T2 1 + (-2.50 - 1.44i)T + (39.5 + 68.4i)T^{2}
83 1+(−1.01−3.79i)T+(−71.8+41.5i)T2 1 + (-1.01 - 3.79i)T + (-71.8 + 41.5i)T^{2}
89 1−12.7T+89T2 1 - 12.7T + 89T^{2}
97 1+(−4.41+7.63i)T+(−48.5−84.0i)T2 1 + (-4.41 + 7.63i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.09922210061007790673499406167, −11.92988062322418391378699187557, −10.83707118560808245556092244499, −10.02876946704864467798938227619, −9.239712029860852839445584968349, −7.996993520319502246536579363900, −6.13134889517005725772449292194, −4.68994559020438820420859011833, −3.74430714208859079142474023266, −2.20246958531668556763664102473, 2.45300818335388549102721879228, 4.21175919832984363082146953782, 5.70253716562732335634887128064, 6.88721959833471951344045816538, 7.67009910976264429980881239450, 8.915127787997026031688704716346, 9.595926219028649090517333322523, 11.93024414738925937920834358571, 12.22272049219969137843581302174, 13.48427112854635897038813299989

Graph of the ZZ-function along the critical line