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Properties

Label 2-12e2-144.131-c1-0-5
Degree 22
Conductor 144144
Sign 0.561−0.827i0.561 - 0.827i
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  + (−0.311 + 1.37i)2-s + (−1.36 − 1.06i)3-s + (−1.80 − 0.858i)4-s + (2.31 − 0.619i)5-s + (1.89 − 1.55i)6-s + (2.51 + 4.35i)7-s + (1.74 − 2.22i)8-s + (0.733 + 2.90i)9-s + (0.135 + 3.38i)10-s + (−1.03 − 0.276i)11-s + (1.55 + 3.09i)12-s + (4.00 − 1.07i)13-s + (−6.78 + 2.11i)14-s + (−3.81 − 1.61i)15-s + (2.52 + 3.10i)16-s − 2.22i·17-s + ⋯
L(s)  = 1  + (−0.219 + 0.975i)2-s + (−0.788 − 0.614i)3-s + (−0.903 − 0.429i)4-s + (1.03 − 0.276i)5-s + (0.773 − 0.634i)6-s + (0.949 + 1.64i)7-s + (0.617 − 0.786i)8-s + (0.244 + 0.969i)9-s + (0.0427 + 1.06i)10-s + (−0.310 − 0.0833i)11-s + (0.448 + 0.893i)12-s + (1.11 − 0.297i)13-s + (−1.81 + 0.564i)14-s + (−0.985 − 0.416i)15-s + (0.631 + 0.775i)16-s − 0.539i·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.561−0.827i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((0.561−0.827i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.561−0.827i0.561 - 0.827i
Analytic conductor: 1.149841.14984
Root analytic conductor: 1.072301.07230
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ144(131,⋅)\chi_{144} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1/2), 0.561−0.827i)(2,\ 144,\ (\ :1/2),\ 0.561 - 0.827i)

Particular Values

L(1)L(1) ≈\approx 0.814223+0.431385i0.814223 + 0.431385i
L(12)L(\frac12) ≈\approx 0.814223+0.431385i0.814223 + 0.431385i
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+(0.311−1.37i)T 1 + (0.311 - 1.37i)T
3 1+(1.36+1.06i)T 1 + (1.36 + 1.06i)T
good5 1+(−2.31+0.619i)T+(4.33−2.5i)T2 1 + (-2.31 + 0.619i)T + (4.33 - 2.5i)T^{2}
7 1+(−2.51−4.35i)T+(−3.5+6.06i)T2 1 + (-2.51 - 4.35i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.03+0.276i)T+(9.52+5.5i)T2 1 + (1.03 + 0.276i)T + (9.52 + 5.5i)T^{2}
13 1+(−4.00+1.07i)T+(11.2−6.5i)T2 1 + (-4.00 + 1.07i)T + (11.2 - 6.5i)T^{2}
17 1+2.22iT−17T2 1 + 2.22iT - 17T^{2}
19 1+(0.697−0.697i)T−19iT2 1 + (0.697 - 0.697i)T - 19iT^{2}
23 1+(2.20+1.27i)T+(11.5+19.9i)T2 1 + (2.20 + 1.27i)T + (11.5 + 19.9i)T^{2}
29 1+(−0.589−0.157i)T+(25.1+14.5i)T2 1 + (-0.589 - 0.157i)T + (25.1 + 14.5i)T^{2}
31 1+(−0.190−0.109i)T+(15.5+26.8i)T2 1 + (-0.190 - 0.109i)T + (15.5 + 26.8i)T^{2}
37 1+(5.16−5.16i)T−37iT2 1 + (5.16 - 5.16i)T - 37iT^{2}
41 1+(−0.828+1.43i)T+(−20.5−35.5i)T2 1 + (-0.828 + 1.43i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.33+4.98i)T+(−37.2−21.5i)T2 1 + (-1.33 + 4.98i)T + (-37.2 - 21.5i)T^{2}
47 1+(5.76+9.98i)T+(−23.5+40.7i)T2 1 + (5.76 + 9.98i)T + (-23.5 + 40.7i)T^{2}
53 1+(7.80+7.80i)T+53iT2 1 + (7.80 + 7.80i)T + 53iT^{2}
59 1+(−1.36−5.09i)T+(−51.0+29.5i)T2 1 + (-1.36 - 5.09i)T + (-51.0 + 29.5i)T^{2}
61 1+(−1.73+6.48i)T+(−52.8−30.5i)T2 1 + (-1.73 + 6.48i)T + (-52.8 - 30.5i)T^{2}
67 1+(2.20+8.22i)T+(−58.0+33.5i)T2 1 + (2.20 + 8.22i)T + (-58.0 + 33.5i)T^{2}
71 1−12.0iT−71T2 1 - 12.0iT - 71T^{2}
73 1−10.3iT−73T2 1 - 10.3iT - 73T^{2}
79 1+(−7.74+4.46i)T+(39.5−68.4i)T2 1 + (-7.74 + 4.46i)T + (39.5 - 68.4i)T^{2}
83 1+(1.48−5.55i)T+(−71.8−41.5i)T2 1 + (1.48 - 5.55i)T + (-71.8 - 41.5i)T^{2}
89 1−6.56T+89T2 1 - 6.56T + 89T^{2}
97 1+(1.51+2.62i)T+(−48.5+84.0i)T2 1 + (1.51 + 2.62i)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

−13.37204637004516186639832527266, −12.42137181706127097685327111899, −11.29975776470268052940488686879, −10.04663877188632171474014025840, −8.767292289752980243200875209820, −8.094190371730043670012922181964, −6.50746821732284909206105843456, −5.62010443532444529864040713584, −5.10199268679097318727039382328, −1.80525785726359857899404836537, 1.44451580822680463203449719550, 3.77942470064949970809569654358, 4.78245994375795329945535886328, 6.21283283513354489766254723161, 7.80680844718232791816409396685, 9.250918391333685031694772746360, 10.34465791025165582490065607032, 10.71364398875723499479179700824, 11.50938584840679905058223200292, 12.93396322858860133676261848423

Graph of the ZZ-function along the critical line