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Properties

Label 2-12e2-144.11-c1-0-9
Degree 22
Conductor 144144
Sign 0.335+0.942i0.335 + 0.942i
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  + (−1.39 + 0.225i)2-s + (−1.36 + 1.05i)3-s + (1.89 − 0.628i)4-s + (−2.78 − 0.746i)5-s + (1.67 − 1.78i)6-s + (1.16 − 2.02i)7-s + (−2.50 + 1.30i)8-s + (0.753 − 2.90i)9-s + (4.05 + 0.415i)10-s + (5.53 − 1.48i)11-s + (−1.93 + 2.87i)12-s + (−3.90 − 1.04i)13-s + (−1.17 + 3.08i)14-s + (4.60 − 1.93i)15-s + (3.20 − 2.38i)16-s − 6.45i·17-s + ⋯
L(s)  = 1  + (−0.987 + 0.159i)2-s + (−0.790 + 0.611i)3-s + (0.949 − 0.314i)4-s + (−1.24 − 0.333i)5-s + (0.683 − 0.730i)6-s + (0.440 − 0.763i)7-s + (−0.887 + 0.461i)8-s + (0.251 − 0.967i)9-s + (1.28 + 0.131i)10-s + (1.67 − 0.447i)11-s + (−0.558 + 0.829i)12-s + (−1.08 − 0.290i)13-s + (−0.313 + 0.824i)14-s + (1.19 − 0.498i)15-s + (0.802 − 0.596i)16-s − 1.56i·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.335+0.942i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((0.335+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.335+0.942i0.335 + 0.942i
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), 0.335+0.942i)(2,\ 144,\ (\ :1/2),\ 0.335 + 0.942i)

Particular Values

L(1)L(1) ≈\approx 0.323110−0.227916i0.323110 - 0.227916i
L(12)L(\frac12) ≈\approx 0.323110−0.227916i0.323110 - 0.227916i
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+(1.39−0.225i)T 1 + (1.39 - 0.225i)T
3 1+(1.36−1.05i)T 1 + (1.36 - 1.05i)T
good5 1+(2.78+0.746i)T+(4.33+2.5i)T2 1 + (2.78 + 0.746i)T + (4.33 + 2.5i)T^{2}
7 1+(−1.16+2.02i)T+(−3.5−6.06i)T2 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2}
11 1+(−5.53+1.48i)T+(9.52−5.5i)T2 1 + (-5.53 + 1.48i)T + (9.52 - 5.5i)T^{2}
13 1+(3.90+1.04i)T+(11.2+6.5i)T2 1 + (3.90 + 1.04i)T + (11.2 + 6.5i)T^{2}
17 1+6.45iT−17T2 1 + 6.45iT - 17T^{2}
19 1+(1.50+1.50i)T+19iT2 1 + (1.50 + 1.50i)T + 19iT^{2}
23 1+(0.0418−0.0241i)T+(11.5−19.9i)T2 1 + (0.0418 - 0.0241i)T + (11.5 - 19.9i)T^{2}
29 1+(5.08−1.36i)T+(25.1−14.5i)T2 1 + (5.08 - 1.36i)T + (25.1 - 14.5i)T^{2}
31 1+(1.65−0.952i)T+(15.5−26.8i)T2 1 + (1.65 - 0.952i)T + (15.5 - 26.8i)T^{2}
37 1+(−0.489−0.489i)T+37iT2 1 + (-0.489 - 0.489i)T + 37iT^{2}
41 1+(−0.0155−0.0269i)T+(−20.5+35.5i)T2 1 + (-0.0155 - 0.0269i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.01+3.80i)T+(−37.2+21.5i)T2 1 + (1.01 + 3.80i)T + (-37.2 + 21.5i)T^{2}
47 1+(−0.0913+0.158i)T+(−23.5−40.7i)T2 1 + (-0.0913 + 0.158i)T + (-23.5 - 40.7i)T^{2}
53 1+(−6.62+6.62i)T−53iT2 1 + (-6.62 + 6.62i)T - 53iT^{2}
59 1+(−1.11+4.15i)T+(−51.0−29.5i)T2 1 + (-1.11 + 4.15i)T + (-51.0 - 29.5i)T^{2}
61 1+(1.71+6.39i)T+(−52.8+30.5i)T2 1 + (1.71 + 6.39i)T + (-52.8 + 30.5i)T^{2}
67 1+(−0.216+0.808i)T+(−58.0−33.5i)T2 1 + (-0.216 + 0.808i)T + (-58.0 - 33.5i)T^{2}
71 1+1.04iT−71T2 1 + 1.04iT - 71T^{2}
73 1−4.74iT−73T2 1 - 4.74iT - 73T^{2}
79 1+(−7.29−4.21i)T+(39.5+68.4i)T2 1 + (-7.29 - 4.21i)T + (39.5 + 68.4i)T^{2}
83 1+(−3.20−11.9i)T+(−71.8+41.5i)T2 1 + (-3.20 - 11.9i)T + (-71.8 + 41.5i)T^{2}
89 1−2.85T+89T2 1 - 2.85T + 89T^{2}
97 1+(−3.29+5.71i)T+(−48.5−84.0i)T2 1 + (-3.29 + 5.71i)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

−12.25400500492880877235881242893, −11.58165520080672154269828111604, −11.03359427719893126090852289885, −9.747838399369550712318145221957, −8.892535742127168504410794089005, −7.53187640281573920414744191821, −6.76285174057287760833573620927, −5.04707143667454557275566780735, −3.76751611932508054265942669089, −0.60872501127766903055681199334, 1.86300766391380955148194981557, 4.07602121879338611532859137857, 6.06162321454245023108288229394, 7.11546558744579052614661711187, 7.931022686598030205387250419296, 9.042537297281798947547978581241, 10.43050443944445927597160885766, 11.53567641687946449563194226079, 11.91140523949891595102773207439, 12.62013631044794417291694662464

Graph of the ZZ-function along the critical line