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Properties

Label 2-12e2-144.11-c1-0-8
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(11,⋅)\chi_{144} (11, \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

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

Graph of the ZZ-function along the critical line